Optimization Problems in Telecommunications with Military Applications

Permanent Link: http://ufdc.ufl.edu/UFE0021140/00001

Material Information

Title: Optimization Problems in Telecommunications with Military Applications
Physical Description: 1 online resource (177 p.)
Language: english
Creator: Commander, Clayton W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007


Subjects / Keywords: combinatorial, heuristics, military, optimization, telecommunications
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation


Abstract: In recent decades, optimization problems in telecommunication systems have been the focus of an intensive amount of research. These problems are important for several reasons including speed and quality of communication among others. In this dissertation, we present several problems arising in telecommunication networks in military applications. Several problems we consider involve wireless communication networks. These networks are an extraordinarily convenient method of communication. However, along with this convenience comes a myriad of complicated problems that must be addressed to preserve the attractive features of the networks. Furthermore, problems arising in adversarial environments differ from those in conventional settings, in that time is usually a critically constrained factor. This is troublesome because many of the problems are difficult to solve and would require a tremendous amount of time to compute the optimal solution. However in a battlespace environment, time spent computing a solution and not fighting the enemy leads to a potential loss of materiel and lives. Thus for the problems studied, we will focus a great deal of attention on designing heuristic algorithms which are capable of computing near optimal solutions very efficiently. We will consider two classes of problems involving telecommunication networks. The first class focuses on denying communication on a network and destroying its functionality. The other class has the objective of guaranteeing communication on a network. At first glance, these two sets appear to be polar opposites of one another. However, with any emerging technology studies which assess both vulnerabilities and capabilities must be performed in order to achieve a system which will not fail in its intended operational environment. Our goal is to show how these problems can be formulated and solved using tools from global and combinatorial optimization. For the problems considered, we examine the computational complexity and examine several mathematical programming formulations. Then we present several algorithms and examine extensive computational results comparing their effectiveness. Finally, we conclude by summarizing our work and indicating future directions of research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Clayton W Commander.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021140:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021140/00001

Material Information

Title: Optimization Problems in Telecommunications with Military Applications
Physical Description: 1 online resource (177 p.)
Language: english
Creator: Commander, Clayton W
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007


Subjects / Keywords: combinatorial, heuristics, military, optimization, telecommunications
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation


Abstract: In recent decades, optimization problems in telecommunication systems have been the focus of an intensive amount of research. These problems are important for several reasons including speed and quality of communication among others. In this dissertation, we present several problems arising in telecommunication networks in military applications. Several problems we consider involve wireless communication networks. These networks are an extraordinarily convenient method of communication. However, along with this convenience comes a myriad of complicated problems that must be addressed to preserve the attractive features of the networks. Furthermore, problems arising in adversarial environments differ from those in conventional settings, in that time is usually a critically constrained factor. This is troublesome because many of the problems are difficult to solve and would require a tremendous amount of time to compute the optimal solution. However in a battlespace environment, time spent computing a solution and not fighting the enemy leads to a potential loss of materiel and lives. Thus for the problems studied, we will focus a great deal of attention on designing heuristic algorithms which are capable of computing near optimal solutions very efficiently. We will consider two classes of problems involving telecommunication networks. The first class focuses on denying communication on a network and destroying its functionality. The other class has the objective of guaranteeing communication on a network. At first glance, these two sets appear to be polar opposites of one another. However, with any emerging technology studies which assess both vulnerabilities and capabilities must be performed in order to achieve a system which will not fail in its intended operational environment. Our goal is to show how these problems can be formulated and solved using tools from global and combinatorial optimization. For the problems considered, we examine the computational complexity and examine several mathematical programming formulations. Then we present several algorithms and examine extensive computational results comparing their effectiveness. Finally, we conclude by summarizing our work and indicating future directions of research.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Clayton W Commander.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Pardalos, Panagote M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021140:00001

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S2007 Clayton WNarren Commander

To my family.


First, I must thank my advisor Panos Pardalos. He has been an incredible mentor, leader, and

friend to me since the day we met. His excitement and passion for life, research, and family have

had a profound effect on me and have encouraged me greatly. H-e will always have a ;.pdl-ll; place

in my heart.

Mly appreciation goes to my committee members, Stan Uryasev, J. Cole Smith, and William

Hager for their time and helpful ideas that guided me along the way. I would also like to thank

the members of the graduate committee Farid AitSahlia, Elif Akgali, and Edwin Romeijn for

giving a student from whom they had nothing to gain another chance.

Next I wvish to thank my wonderful co-authors and collaborators for working with me

and for helping me learn how to be a researcher: Ashwin Arulselvan, Sergiy Butenko, Lily

Elefteriadou, Paola Festa, Michael J. Hirsch, Carlos A.S. Oliveira, Michelle Ragle, Mauricio

G.C. Resende, Valeriy Ryabchenko, Oleg Shylo, Marco Tsitselis, Stan Uryasev, Yinyu Ye,

and Grigory Zharshevsky. I would like to thank Jonathan King for instilling in me a love of

mathematics and an appreciation for clearly written mathematical discourse. I am particularly

grateful to Mauricio G.C. Resende of AT&T Labs Research for his wonderful collaborations and

help over the years. He is every bit as gracious as he is brilliant. Finally, I am grateful to Claudio

Meneses and Onur Seref for their thoughtful advice.

I am truly grateful to the United States Air Force for supporting and financing my

educational endeavors. Particular thanks go to Rob Murphey, David Jeffcoat, Michelle White,

and many others at the Air Force Research Laboratory for their support. Thanks go to Don

Grnmdel who always gave good advice and kept me on the straight and narrow.

Finally, but certainly not least, my most heartfelt appreciation goes to my family. I thank my

parents and my grandmother who always listened and encouraged me. I thank my parents-in-law

for always giving me a place to stay. Finally, I thank my beautiful wife Leah who has been my

constant source of love, passion, and inspiration.






ABSTRACT. ................................ . 13




2.1 Introduction . . 16
2.2 Idiosyncrasies . . 16
2.3 Fundamental Results . . 17
2.4 Discrete Optimization . . 21
2.5 Computational Complexity . . 22
2.6 Upper and LowerBounds . . 24
2.7 Algorithms for Optimization Problems . . 28
2.7.1 Exact Methods. . . 28
2.7.2 Heuristics . . 31
2.8 Concluding Remarks . . 36


3.1 Introduction . . 38
3.2 Problem Formulations. . . 40
3.2.1 Critical Node Problem. . . 40
3.2.2 Cardinality Constrained Problem . . 44
3.3 Heuristics for Critical Node Problems . . 46
3.3.1 CNP Heuristic . . 46
3.3.2 CC-CNP Heuristic. . . . 49
3.3.3 Genetic Algorithm for the CC-CNP . . 50
3.4 Computational Results . . 53
3.4.1 CNP Results . . 53
3.4.2 CC-CNP Results. . . 55
3.5 Concluding Remarks . . 59


4.1 Introduction . . 62
4.2 Definitions and Assumptions . . 63

4.3 Deterministic Formulations .
4.3.1 Coverage Approach
4.3.2 Connectivity Formulation .
4.4 Deterministic Setup with Percentile Constraints
4.4.1 Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) .
4.4.2 Percentile Constraints and the WNJP .
4.5 Case Studies and Algorithms
4.5.1 Coverage Formulation. .
4.5.2 Connectivity Formulation .
4.6 Concluding Remarks


Descriptions, Assumptions, and Definitions
Problem Formulation
Upper and Lower Bounds.
Heuristic for Uncertain Jamming.
Concluding Remarks



6.1 Introduction . ............... ........ . 106
6.2 Discrete Formulations (CCPMANET-D) . . 109
6.3 Algorithms for CCPMANET-D . . 113
6.3.1 Construction Heuristic . . 113
6.3.2 Local Improvement . . 115
6.3.3 One-Pass Heuristic . . 117
6.3.4 Greedy Randomized Adaptive Search . . 119
6.3.5 Complexity of the Heuristic . . 123
6.3.6 Computational Experiments . . 125
6.4 A Continuous Formulation (CCPMANET-C) . . 131
6.5 Concluding Remarks . . 136



Introduction . . 139
Problem Description . . 140
Computational Complexity . . 142
H euristics . . . . ... ... .. ... .. . .145
7.4.1 Combinatorial Algorithm for TDMA Message Scheduling . . 145
7.4.2 G R A SP . . . . . ... .. . 150
7.4.3 Sequential Vertex Coloring . . 151
7.4.4 Mean Field Annealing . . 152
7.4.5 Mixed Neural-Genetic Algorithm . . 152

7.5 Computational Results . . 153
7.6 Concluding Remarks . . 159

8 CONCLUSION . ........................... .. .162




Table page

2-1 Growth rates of several polynomial and non-polynomial functions. . . 24

3-1 Results of IP model and heuristic on terrorist network data. . . 53

3-2 Results of IP model and heuristic on randomly generated scale free graphs. . 56

3-3 Results of IP model and heuristics on terrorist network data. . . 57

3-4 Results of the IP model and genetic algorithm and the combinatorial heuristic on
randomly generated scale free graphs. . . 58

3-5 Comparative results of the genetic algorithm and the combinatorial heuristic when
tested on the larger random graphs. Due to the complexity, we were unable to
compute the corresponding optimal solutions. . . 59

4-1 Optimal solutions using the coverage formulation with regular and VaR constraints. 80

4-2 Optimal solutions using the coverage formulation with regular and VaR, and CVaR
constraints. . . 81

5-1 Comparing -for various values of k. .. ........ .. 95

5-2 Numerical results are provided for several regions with various required j amming
levels. The Upper Bound, Lower Bound, Optimal Grid, and Local Search columns
provide the number of jamming devices required for the corresponding region
according to the theorems presented and the proposed local search. The Percent
Decrease shows the savings when comparing the local search to the optimal grid
approach . . 105

6-1 Comparative results between shortest path solutions and heuristic solutions. . 118

6-2 Three instances with different sets of agents on 50 node graphs are given. The value
in the UBound column was found using Corollary 1........... . . 127

6-3 Three instances with different sets of agents on 75 node graphs are given. The value
in the UBound column was found using Corollary 1........... . . 128

6-4 A 100 node instance with solutions with radius varying from 1 to 5 units. The value
in UBound was found using Corollary 1................. . . 129

6-5 Average solution values for GRASP and GRASP with path-relinking on 50 node
graphs.. ... ... .. ... ... ... ... .. ... ... .... . .130

6-6 Comparative solutions of GRASP and GRASP with path-relinking on 75 node graphs. 130

6-7 Results of GRASP and GRASP with path-relinking on 100 node graphs. . . 131

7-1 Comparison of solutions for the benchmark instances from Wang and Ansari. . 156

7-2 Comparison of optimal and heuristic solutions for graphs with | V| = 50 stations. An *
indicates that the solution is optimal, while a t indicates the solution is the best found by
Xpress-MP after 3600s. Solutions are reported as (X,1) M) ..... . . 157

7-3 Comparison of optimal solver and heuristic solutions for the 75 station networks. . 158

7-4 Comparison of optimal solver and heuristic solutions for networks with |V| = 100 stations. 160


Figure page

2-1 Notice that the rounded integer solution is not optimal. . . 22

2-2 Visualization of complexity classes. . . 23

2-3 Pseudo-code for a greedy algorithm which makes change using the minimum number
of coins. . . 32

2-4 GRASP for maximization . . 33

2-5 Generic simulated annealing maximization algorithm. . . 35

2-6 Pseudo-code for generic genetic algorithm.. . . 36

3-1 Connectivity Index of nodes A,B,C,D is 3. Connectivity Index of E,F,G is 2.
Connectivity Index of H is 0. ....................... . . 45

3-2 Heuristic for detecting critical nodes. . . 46

3-3 Local search algorithm for critical node heuristic. . . 48

3-4 Heuristic with local search for detecting critical nodes. . . 49


3-6 Pseudo-code for generic genetic algorithm. . . 51

3-7 Example of the crossover operation. In this case, CrossProb = 0.65. . . 52

3-8 Terrorist network compiled by Krebs. . . 54

3-9 Optimal solution when k = 20. . . 55

3-10 Optimal solution when L = 4. . . 57

4-1 Connectivity Index of nodes A,B,C,D is 3. Connectivity Index of E,F,G is 2.
Connectivity Index of H is 0. ....................... . . 66

4-2 Graphical representation of Var and CVaR. . . 73

4-3 Case study 1. The placement of jammers is shown when the problem is solved using
the original and VaRconstraints. . . 81

4-4 Case study 1 continued. The placement of jammers is shown when the problem is
solved using VaR and CVaR constraints. . . 82

4-5 Case Study 2: Original graph. . . 83

4-6 A comparison of the percentile constrained solutions. In both cases, the triangles
represent the placement of jamming devices. . . 83

Uniform grid with j amming devices ....

The least covered point is shown in the lower left grid cell. ....

Square Decomposition ....

Equivalent Points ....

Cumulative emanation of jamming devices. ....

Integral Lower Bound. ....

Integral Upper Bound. ....

Comparison of the lower and upper bounds. ....

Pseudo-code for the randomized local search for uncertain j amming. .

0 Example of heuristic versus uniform placement. ....

Pseudo-code for the shortest-path construction heuristic. ....

Pseudo-code for the Hill Climbing intensification procedure. ....

Pseudo-code for the one-pass heuristic. ....

GRASP for maximization ....

Greedy randomized constructor for CCPMANET-D. ....

Local search for CCPMANET-D. ....

Path-relinking subroutine. .....

GRASP with path-relinking for maximization. ....

. 88

. 89

. 89

. 90

. 91

. 92

. 97

. .100

. 103



. 116

. 117

. 119



.. ...124

. .126

. .132

. 133




. .137

6-9 Evolution of GRASP+PR solution values on
radius increases from 1 to 5 units. ....

6-10 Evolution of GRASP+PR solution values on
radius increases from 1 to 5 units. ....

6-11 Evolution of GRASP+PR solution values on
radius increases from 1 to 5 units. ....

6-12 The heavyside function H1. ....

6-13 Alternate objective function H2. ....

6-14 Second alternate objective function H3. .

50 node graphs as the com

75 node graphs as the com

100 node graphs as the cor

7-1 Counterexample to the claim of Wang & Ansari that optimal graph coloring can be
found by recursively finding a maximum independent set and removing it from the
graph. . . . . . . . . ... . .143

7-2 Construction of graph G' from G. . . 144

7-3 Pseudo-code of the proposed heuristic for MSP-TDMA. . . 146

7-4 Greedy randomized heuristic for frame length minimization. . . 147

7-5 Throughput maximization pseudo-code. . . 149

7-6 Benchmark TDMA test cases. . . 155

7-7 Example GRASP broadcast schedules for the networks given in Figure 7-6: (a) 15
station network, (b) 30 station network, (c) 40 station network. . . 161

Abstract of Dissertation Presented to the G~raduate School
of the University of Florida in Partial Fulfillment of the
t;,ii lir. com, of -. for the Degree of Doctor of Philosophy



Clayton WVarren Commander

August 2007

Chair: Panagote M. Pardalos
N T~a.->-r-: Industrial and Systems Engineering

In recent decades, optimization problems in telecommunication systems have been the focus

of an intensive amount of research. These problems are important for several reasons including

speed and quality of communication among others. In this dissertation, we present several

problems arising in telecommunication networks in military applications. Several problems

we consider involve wMireless communication networks. These networks are an extraordinarily

convenient method of communication. H-ow~ever, along with this convenience comes a myriad of

complicated problems that must be addressed to preserve the attractive features of the networks.

Furthermore, problems arising in adversarial environments differ from those in conventional

settings, in that time is I .1.1. ll; a critically constrained factor. This is troublesome because many

of the problems are difficult to solve and would require a tremendous amount of time to compute

the optimal solution. HiowYever in a b-. iri.ln--:.rl- environment, time spent computing a solution

and not fighting the enemy leads to a potential loss of materiel and lives. Thus for the problems

studied, we will focus a great deal of attention on designing heuristic algorithms which are

capable of c-;i p Itint a.I near optimal solutions very efficiently.

Wie will consider two classes of problems involving telecommunication networks. The first

class focuses on denying communication on a network and destroying its functionality. The

other class has the objective of guaranteeing communication on a network. At first glance, these

twio sets appear to be polar opposites of one another. However, with any emerging technology

studies which assess both vulnerabilities and capabilities must be performed in order to achieve

a system which will not fail in its intended operational environment. Our goal is to showN

how these problems can be formulated and solved using tools from global and combinatorial

optimization. For the problems considered, we examine the computational complexity and

examine several mathematical programming formulations. Then we present several algorithms

and examine extensive computational results ca~lll~ onesina their effectiveness. Finally, we conclude

by ; Iin lisl m; ;lin our work and indicating future directions of research.


Optimization problems in telecommunication systems have been the focus of an

intensive amount of research in recent decades [135, 155]. These problems are important

for several reasons including speed and quality of communication and cost related issues.

In this dissertation, we present several problems arising in military applications involving

telecommunication networks. Several problems we consider involve wireless networks. These

networks are an extraordinarily convenient method of communication; however, along with

this convenience comes a myriad of complicated problems which must be addressed in order

to preserve the attractive features of the networks. Furthermore, problems arising in adversarial

environments differ from conventional problems in that time is usually a critically constrained

factor. This presents somewhat of a problem because many of the problems are extremely

difficult to solve and would require a tremendous amount of time to compute the optimal

solution. However in a battlespace environment, time spent computing a solution and not fighting

the enemy leads to a potential loss of materiel and lives. Thus throughout this dissertation we will

focus a great deal of attention on designing heuristic algorithms which are capable of computing

near optimal solutions very efficiently.

The remaining chapters of this dissertation present the results of my efforts to model and

solve many important telecommunication problems facing the military in the ever evolving

global war on terrorism. We will consider two classes of problems involving telecommunication

networks. The first class (Chapters 3, 4 and 5) focus on denying communication on a network

and destroying its functionality. Conversely, the problems in Chapter 6 and Chapter 7 have the

obj ective of guaranteeing communication on a network. At first glance, these two sets appear to

be polar opposites of one another. However, with any emerging technology studies which assess

both vulnerabilities and capabilities must be performed to achieve a system which will not fail in

its intended operational environment. Our goal is to show how these problems can be formulated

and solved using tools from global and combinatorial optimization [74, 106, 127].


2.1 Introduction

Over the past 60 years, Operations Research (OR) has emerged as one of the most exciting,

fast-paced, and interdisciplinary fields of mathematics. Since its rebirth during World War II,

OR has turned into a fascinating subj ect which crosses all divides from real analysis, probability,

statistics, economics, theoretical computer science, and biology in an attempt to solve some of

the most computationally difficult problems known to exist.

As mentioned in [98], OR was first formalized during World War II when supplies were

limited and needed to be allocated to the allied forces overseas. OR teams were fundamental in

developing methods for using radar which was crucial in the allies winning the air war. Later,

researchers developed methods for optimally transporting convoys and derived methods for

tracking submarines thus leading to success in the Pacific theater. The original name of the field

was M~ilitary Operations Research; however, due to the success of the methods derived during the

war, scientists and engineers began applying these techniques to other problems in mathematics

and industrial engineering. The word military was subsequently dropped because of this.

Since the early 1950's, researchers have been expanding the techniques and methods of OR.

Reminiscent of the time of Gauss and Euler, scientists are making contributions at incredible

rates in fields ranging from facility location problems to the mapping of the human genome.

With the advent of the digital computer, algorithms are now able to be implemented providing

the capability to solve problems never before thought tractable. In this chapter we present the

foundation of global optimization. This will provide the necessary tools for us as we investigate

the problems presented in the succeeding chapters.

2.2 Idiosyncrasies

In this subsection, we introduce the symbols and notations we will employ most frequently

throughout this dissertation. Denote a graph G = (V, E) as a pair consisting of a set of vertices

V, and a set of edges E. Let the map w : E H R be a weight function defined on the set of

edges. We will denote an edge-weighted graph as a pair (G, w). Thus we can easily generalize

an un-weighted graph G = (V, E) as an edge-weighted graph (G, w), by defining the weight

function as

0, if (i, j) 5' E. 21

We use the symbol "b := a" to mean "the expression a defines the (new) symbol b" in the

manner of King [115]. Of course, this could be conveniently extended so that a statement like

"(1 e)/2 := 7" means "define the symbol e so that (1 e)/2 = 7 holds" [114]. We will

employ the typical symbol S" to denote the complement of the set S; further let A UB denote the

set-difference, A n Be. Agree to let the expression x <- y mean that the value of the variable y

is assigned to the variable x. To denote the cardinality of a set S, we use | S|. Finally, we will use

italics for emphasis, and SMALL CAPS for formal problem names. Any other locally used terms

and symbols will be defined in the sections in which they appear.

2.3 Fundamental Results

In global optimization, the obj ective is to determine the maximum or minimum point

attained by an obj ective function defined over a set. In general, an optimization problem has the


minimize or maximize f (x)

subject to x E S,

where SC cR" is the feasible region and f (x) is a real valued function defined on S. That is,

f : SH R'.

Definition 1. An optimization problem aI ith feasible region S C IR" is said to be infeasible if

Throughout this dissertation, we will rely heavily on the notion of a neighborhood which is

defined next.

Definition 2. For a given optimization problem on a set S C I-: a neighborhood is a mapping

NV: S H2s

defined for each instance.

In subsequent chapters we will see that cleverly defining a neighborhood for a particular

problem can greatly increase the effectiveness of heuristics. For example, if S = IR", then the set

of points that fall within some Euclidean distance provide a natural choice for the neighborhood

[144]. If || || represents the Euclidean norm, then a point x* E S is said to be a local minimum

point of f if f(x*) < f(x) for all points x E S such that ||Ix x*| < e for some e > 0. In other

words, given e > 0, define the neighborhood of x* as

rv(x*) := {x : x E S and ||x x*| < e} ~. (2-2)

Then, x* is a local minimum if f (x*) < f (x) for all x E 1V(x*). A point x* is said to be a global

minimum if f (x*) < f (x), for all x E Sn {x : ||Ix x*| < e} ~ [107]. The global minimum point

is referred to as an optimal solution. As we will see in the upcoming theorems, the existence of

local minima can sometimes lead to incredible difficulties when searching for a global minimum.

We will later show that in certain cases it is a very hard problem to determine if a local minimum

is also a global minimum.

Before we investigate the fundamental results of global optimization, we provide some basic


Definition 3. A function f : I H D, is continuous at a point c ElIc R if given any e > 0, there

exists 6 > 0 such that if x ElI and |x c| < 6, then | f(x) f (c) | < e.

Definition 4. A set S C IR" is said to be closed ifS contains the limits of all convergent

sequences ofpoints xi E S.

Definition 5. A set S C IR" is said to be compact ifS is both closed and bounded.

We know from real analysis that if S is a compact set, then every infinite sequence of points

xi E S has a convergent subsequence whose limit is in S [161]. Furthermore, for a continuous

function f f (xi) f (x*) whenever xi x*. This leads us to fundamental result by

Weierstrass which we state without proof [107].

Theorem 1. If S is a nonempty compact set in IR", and f (x) is a continuous function defined on

S, then f (x) has at least one global minimum (maximum) point in S.

We can now move on and examine some properties of local and global minima. Recall from

calculus that if the function f is continuously differentiable in a neighborhood of a point x* E S,

and de ER", then dTV f(x*) is said to be the directional derivative of f at x* in the direction d.

If we fix x* and d, then the function h(X) := f (x* + Ad), for As ER+, describes f along the ray

{x = x* + Ad, A > 0}. If we evaluate the derivative of h with respect to A at the point A = 0,

using the first order Taylor expansion of h at A, we see that this is precisely d V f(x*). Thus,

d V f(x*) < 0 implies that there exists A > 0 such that f (x* + Ad) < f (x*) for every 0 < A < A.

Hence the condition d V f(x*) < 0 put simply, implies that we can decrease f locally in the

direction d. Clearly, we are only interested in those directions d that do not immediately leave the

feasible region S.

Definition 6. A direction vector de ER" is said to be a feasible direction at x* if there exists

A > 0 such that x* + Ad E S r every 0 < A < A*.

This leads us to the following theorem which provides a necessary condition for locally

optimal solutions.

Theorem 2. Suppose that the function f (x) is continuously di~ff~erentiable on an open set

containing S C IR". If x* is a local minimum of f (nl ithr respect to S), then d V f(x*) > 0 for

every d EZ (x*), where Z (x*) is the set of all feasible directions at x*.

x* E S is called a critical point if d V f(x*) > 0 for every d E Z(x*). Now, we will

determine in which cases critical points represent global optima. We first need to recall the

definitions of convex sets and convex functions.

Definition 7. A set S C R is called convex set if for every xl, x2 E S, and A E IR, O < A < 1,

the point Axl + (1 A)x2 E S.

In a geometric sense, S is convex if for any two points in S, the line segment joining these

two points is wholly contained in S [148].

Definition 8. Given a convex set S C IR", the unction : S H R is said to be a convex function

if for any xl, x2 E S, and A E IR, O < A < 1, the following condition holds:

f (Axx + (1 X~A)2 1 2fx) -1 ~fx

The function f is said to be a concave function if and only if f is convex. In the following

theorem, we prove that for optimization problems where f is convex and S is a convex set, that

critical points are always globally optimal solutions.

Theorem 3. Let f : S H R be a convex function, where SC cR", is a convex set. Thzen every

local minimum of f is also a global minimum.

Proof: Let x* be a local minimum point and assume for the sake of contradiction that there exists

another point E S such that f (x) < f (x*). From Definition 8, we know that convexity of f

implies that for any A E R such that 0 < A < 1,

f (x* + A(x x*)) < Af (x) + (1 A) f(x*) < f (x*).

Notice that we have contradicted the local optimality assumption of x* since for a local minimum

point, there must exist some A* > 0 such that for 0 < A < A*, f (x* + A(x x*) > f (x*). O

In the following theorem, we provide an optimality criterion for minimizing a concave

function over a convex set.

Theorem 4. Given a compact convex set Sc CR", the global minimum of a concave function

f : S ~ R is attained at an extreme point of S.

Proof: We know that any point x in a convex set can be written as the convex combination of the

extreme points of S. That is, x = CE zA such that CE As = 1, As > 0 V i. Since f is

concave, we have

f .r > Asf r) A -mi f(r) i=1, N} (2-3)
i= 1 i= 1
= min f (l ) :i= 1,...,N}). (2-4)

Recall that linear functions are both convex and concave. Therefore if we are considering

a limearprogrananzing problem, i.e. that of minimizing a continuous linear function over a

polytope, both Theorem 3 providing optimality of local minima for convex functions and

Theorem 4 providing extreme point optimality of concave functions apply. Thus, for this class of

problems we can restrict the search for the global solution by examining only the extreme points

of the polytope. In the absence of convexity however, a global minimum point can occur at a

point other than an extreme point.

In this dissertation, we focus on problems of this type. In particular the problems we will

later investigate contain many locally optimal solutions which differ from the global solution.

Also, until now we have focused on theorems for continuous functions. However, several

problems we will encounter are have discrete variables. These problems are called conabinatorial

optimiza~tionproblems. The next section contains some basic results regarding combinatorial

problems which we will later use.

2.4 Discrete Optimization

In certain applications, it is necessary to restrict the values of the decision variables of a

problem to be integer valued. Such problems are referred to as integer progrananzing problems.

Sometimes, it is convenient to include integer variables in a problem when one is attempting to

model a situation that has two possible values. In this case, binary variables that take the value 0

or 1 are used.

Integer programming problems present unique challenges in that the techniques and

theorems for linear programming problems as described above do not necessarily apply. For

example, consider the polytope in Figure 2-1. Notice that the integer points do not lie at the

** Optimall LP solution

Optimal IP solution

i~rcinodecreasing cost.

Figure 2-1: Notice that the rounded integer solution is not optimal.

extreme points of the polytope. We see then that the result from Theorem 4 does not hold.

Another common misconception is that the integer optimal solution can be found by rounding

the linear programming solution to the nearest integer. We see by examining the figure that this

doesn't work either. Notice that the integer point nearest the linear programming optimal solution

falls outside the feasible region of the polytope.

Clearly, we need more advanced methods for solving such problems. We will look at

a variety of exact and heuristic methods in Section 2.7. Now, we provide an introduction to

computational complexity. Complexity theory lies at the heart of global optimization and

provides tools for empirically determining the level of difficulty of a given problem as well as the

effectiveness of an algorithm. Later we will confirm our suspicions about the difficulty of integer

programs and nonconvex continuous programming problems.

2.5 Computational Complexity

In this section, we develop a means by which we can classify a problem as either being

"easy" or "hard". Then for the so-called "hard" problems, we look at ways to answer the

question: "how hard' is hard? ".

An algorithm is said to be a polynomial time algorithm if its number of elementary

operations, i.e. its running time on a computer, are in the worst cast, bounded above by a

Figure 2-2: Visualization of complexity classes.

polynomial in the size of the input [107]. For instance, an algorithm is said to be O(F~) if the

polynomial which bounds the running time is of order p in the size of the input data I. An

algorithm is said to be an exponential time algorithm if it is not bounded by a polynomial in the

length of the input [6].

When discussing problems in OR, we split the collection of all problems into two classes as

visualized in Figure 2-2. Those problems which can be solved optimally by a polynomial time

algorithm are said to belong to the class P. The other complexity class contains those problems

which can be solved by a "nondeterministic" algorithm in polynomial time. This class is called

HiP. A problem in MiP is one in which it is easy to verify the correctness of solution, but is very

hard to solve, where a problem belonging to p is simply "easy" to solve.

Among the problems in HiP, those which are the most difficult to solve are said to be

HiP-complete. A problem atl is said to be polynomially transformable to problem c~fag

polynomial time algorithm for ca2 WOuld imply a polynomial time algorithm for atl. Problems

in NiP-complete are special in that every problem in HiP can be polynomially transformed to

every other problem in NiP-complete. Thus, since P C Nip, it follows that if one could design

a polynomial time algorithm for a single HiP-complete problem, then every problem in MiP

could be solved with a polynomial time algorithm and thus p would equal HiP [79]. However,

Table 2-1: Growth rates of several polynomial and non-polynomial functions.
n nl n2 n4 2" n!
10 101 102 104 103 3.6 x106
100 102 104 10s 1.27 x1030 9.33 x10157
1000 103 106 1012 1.07 x10301 4.02 x102,567
10, 000 104 10s 1016 0.99 x103,010 2.85 x1035,659

despite the incredible amount of research and investigation, the question as to whether P = NiP

remains the single greatest unsolved problem in theoretical computer science [145]. In fact, the

Clay Mathematics Institute has named this problem as one of seven prize millennium problems

and is offering $1 million to anyone who presents an answer to the question, "does P = HiP?"


Definition 9. An optimization problem xr is said to be HiP-hard, if there exists and HiP-complete

problem which is polynomially transfr mablert~r~rt~t~rtrt~ to ;T.

Throughout this dissertation, we are going to focus on problems that are HiP-hard and

HiP-complete. The next reasonable question one asks of H~P problems is what the implication is

for solving them. That is, how does being in MiP really complicate the computational tractability

of a problem. Table 2-1 provides a several examples of the growth rates of some polynomial

and exponential functions [6]. Notice how quickly the exponential algorithms grow. This is one

reason why polynomial time algorithms are preferred over exponential time algorithms. Most

discrete optimization problems turn out to be either HiP-hard or HiP-complete, even if they are

linear [149].

2.6 Upper and Lower Bounds

When attempting to solve integer programs (IPs), we are faced with the problem of how to

prove that a given point is an optimal solution [173]. This problem arises since local optimality

does not imply global optimality for IPs. Oftentimes being able to derive upper and lower bounds

on the optimal solution is helpful to identify good approximate solutions and narrow the search

for the optimal solution. This topic will be studied extensively in Chapter 5. Now we introduce

some basic properties of bounds for integer programming problems.

Consider the IP given below and assume that the point x* is an optimal solution.

minimize cx (2-5)

subj ect to x E S (2-6)

x E Z". (2-7)

Then in order to solve this IP we need to determine a lower bound x such that c(x) < c(x*) and

an upper bound x where c(x) > c(x*) such that

c(X) = c(X*) = c(X). (2-8)

In order find these bounds in practice, we need an algorithm that can compute a decreasing

sequence of upper bounds

c(X,) > c(X2) > .. > C Us) > c(X*), (2-9)

and a corresponding increasing sequence of lower bounds

c(X1) < c(X2) < .. < C gt) I C Z*), (2-10)

which stops when

c(xs) c(Ze) < e, (2-11)

for some e > 0 [173].

We use the idea of a relaxation in order to find these bounds. A relaxation typically enlarges

the set of feasible solutions, but is easier to solve than the original problem.

Definition 10. A problem (RP) z" = max f (x) : x E TC IR"} is said' to be a relaxation of (IP)

z = max~c(x) : x E Sc CR"} if

(i) Sc T, and'

(ii) f (x) > c(x) for all x E S.

Using this definition, the following lemma holds.

Lemma 1. If (RP) is a relaxation of (IP), then z" > z.

Proof: Let x* E S be an optimal solution for (IP). Then, we have x* E Sc T, which

implies c(x*) < f (x*). Furthermore, since x* E T, f (x*) is a lower bound on z That is,

z= c(X*) < f (X*) < z and we have the lemma. O

The problem of formulating useful relaxations is an important problem in its own right

which has been studied since the founding of OR [53]. Among the most common relaxations

are the linear programming relaxation, and the Lagrangian relaxation. We now provide a brief

introduction to these.

Definition 11. Given an integer program (IP) z := max~cx : x E Sn Z" c IR" }, the linear

programming relaxation of(IP) is the linear program (LPR) z := max(cx : x E Sc CR"}.

Proposition 1. The linearprogramming problem (LPR) is a relaxation of (IP).

Proof: The proof is trivial as S n Z" C S and the objective function of (LPR) remains the same

as in (IP). Thus, by Lemma 1, we have the result. O

We see then that all linear programming relaxations provide bounds on the original integer

program. Further, the following lemma shows that relaxations can be helpful for identifying cases

in which the original integer program is infeasible [173].

Lemma 2. Given an integer program (IP) z"P := max~cx : x E Sn Z" CE }- and its

correspondingLP relaxation (LPR) z := max(cx : x E Sc CR"}, thefollowingsta~tements


(i) If (LPR) is infeasible, then (IP) is also infeasible.

(ii) If x* is an optimal solution to (LPR) such that x* E Sn Z" and f (x*) = c(x*), then x* is

an optimal solution for (IP).

Proof: (i) This follows from the fact that S n Z" c S. Since (LPR) is infeasible, we have that

S = 0, thus implying S n Z" = 0.

(ii) Since (LPR) is a relaxation, then by Definition 10, x* E Sn Z" implies z"P > c(x*) =

f(x*) = z F. However, by definition z"P < zL implying c(x*) = z"P = zL

Another common relaxation used to tackle hard integer programs is the Lagrangian

relaxation. This method was first introduced by Held and Karp in [96, 97] in a formulation for the

TRAVELING SALESMAN PROBLEM [54]. Lagrangian relaxation relaxes the constraints by adding

them to the obj ective function with an associated penalty. Consider the following optimization


(P) x* = max cr (2-12)

s.t. A4: < b (2-13)

xrE X. (2-14)

Then the corresponding Lagrangian relaxation is formulated as follows.

(P~f )) L (f ) = max Cr + <(b ,4:r (2-15)

xrE X. (2-16)

With this we can formulate the following lemma regarding Lagrangian relaxation.

Lemma 3. (P~ft) is a relaxation o~f (P).

Pr~oof In order for (P0ft)) to be a relaxation of (P), as defined by Definition 10, we must show

the following two conditions: (i) the feasible region of the original problem is a subset of the

relaxed problem, and (ii) for all vectors ts > 0, L(ft) > x*.

Condition (i) follows trivially. To show condition (ii), let :r* E X be an optimal solution

for (P). Then r* is clearly feasible for (P(ft)). Further, since r* feasible for (P), b Azr* > 0.

Therefore cr* < cr* + p(b Axr*), for all real vectors s '> 0. O

From this lemma, we see that any feasible solution to (P(ft)) is an upper bound on the

optimal value of (P). The problem of finding the best, or tightest bound, say L* is known as the

LAGRANGIAN MULTIPLIER PROBLEM [124] and is given as L* := min{L(ft) : 4< > 0}. With

this the following lemma, which is stated without proof, holds.

Lemma 4. For all real vectors p, cx < z* < L* < L(p). Furthermore, if L(p) = L* =

z* = cx*, then x* is optimal for the original problem (P), and p is optimal for the LAGRANGIAN


Linear programming and Lagrangian relaxations are helpful for showing when a solution is

close to (in some cases equals) the optimal solution. They are also useful in branch and bound

algorithms, which we will introduce in the following subsection.

2.7 Algorithms for Optimization Problems

Consider a discrete optimization problem that has a binary decision variables. Since there

are a finite number of integer feasible solutions, in theory one could enumerate all possible

solutions. However, to do this would require 2" function evaluations [31]. This is impractical

since if a > 1000, then with the present computers available the computation time required for

this enumeration would take millions of years. Clearly we need more efficient algorithms to solve

these problems.

Algorithms for optimization problems are broken up into two categories: exact methods

and heuristics. Exact methods guarantee that the termination of the algorithm will result in

the optimal solution provided one exists. Heuristics on the other hand, have no guarantee of

optimality but usually find high quality solutions much faster than the exact methods. In fact,

most nontrivial instances of problems in HiP cannot be solved by exact algorithms. Thus, we

need efficient heuristics to find near optimal solutions to real-world instances. In the following

sub sections, we provide an overview of several exact and heuristic methods that we will use for

solving the problems appearing in later chapters.

2.7.1 Exact Methods

Linear Programming Techniques. We begin our discussion of exact algorithms with the

simplex method for linear programming [52]. Consider an instance of a linear programming


minimize ex (2-17)

subj ect to Ax = b (2-18)

x > 0, (2-19)

where A is an m x n matrix with rank m, c an n-vector, and bT is an m-vector. The simplex

method is an algorithm which moves along the extreme points of the polytope defined by

Ax = b in search of the optimal solution. The extreme points are visited in such a way that the

obj ective function value at a new point is at least as good as the previous. Since we showed in

the previous theorems that the optimal solution to a linear program is an extreme point of the

polytope, the simplex method is guaranteed to find the optimal solution. Notice however, that

this algorithm is not polynomial. The polytope in question can have ( ) extreme points, and it

is possible to construct examples for which the simplex method must enumerate all (") of them.

However, despite the theoretical exponential worst-case complexity of the simplex method, it

is very efficient in practice and is easy to implement. This is not to say that linear programs are

HiP-hard. In fact, all linear programs are in P. The class of algorithms known as interior point

methods are able to solve linear programming problems in polynomial time. The first efficient

interior point algorithm was proposed in 1984 by Karmarkar [111]. The first implementation of

Karmarkar's algorithm was reported in 1991 by Adler, Karmarkar, Resende, and Veiga in [4]

and [5]. Excellent reference texts on linear programming include the work of Chvatal [34] and

Bazaraa et al. [15].

Integer Programming Algorithms. Branch and bound (B&B) algorithms are the most

common class of algorithms used for solving discrete optimization problems [61]. B&B methods

are implicit enumeration techniques based on the idea of divide and conquer [90]. In a B&B

algorithm, the set of feasible solutions is decomposed into smaller and smaller sets until the

optimal solution is eventually reached.

Consider the integer programming problem z"P := max~cx : x E Sn Z" CE }- To

apply the B&B algorithm, the linear programming relaxation z"P := max(cx : x E S} is solved,

generally resulting in a nonintegral solution. This solution is taken as initial upper bound on the

optimal solution. Suppose that in the LP relaxation, some variable xi = xi 5/ Z. Then one way to

branch is to divide the feasible region S into two subdomains, namely

S. :=Sn {x : Xi < [Xs] } (2-20)

S2 := Sn {x : Xi > [Xs] } (2-21)

Notice that S1 U S2 = 0 and S1 n S2 = 0. The two linear programs z"P := max {cx : x E St } and

zL := max~cx : x E S2) are HOW SOlVed and the smallest objective value is taken as the new

upper bound. In essence, a search tree is formed by the repetition of the decomposition/bounding

process applied to each sub-problem. However, due to a pre-established lower bound, many of

the resulting subproblems are pruned from the search tree and not considered. Thus, an optimal

permutation is constructed iteratively, one element at a time [151]. The process is repeated on

variables which are nonintegral until eventually the integer optimal solution z"P is reached [173].

Though branch and bound methods are the most commonly used algorithms for discrete

optimization problems, they are not the only techniques available. The branch and cut method

is a hybrid of branch and bound which falls into the class of so-called cutting plane techniques

[54, 107, 173]. Introduced by Gomory in [88], cutting plane methods solve IPs by introducing

constraints which cuts off the noninteger solution found by solving the LP relaxation without

removing any feasible integer solutions. Finally, column generation, or so-called branch and price

algorithms are effective decomposition methods and are commonly used for solving large-scale

integer programming problems [55, 57, 58]. The design of software packages which efficiently

execute optimal integer programming algorithms is a global enterprise. Today the most efficient

and widely used commercial IP solvers are CPLEXB by ILOG, Inc. [50] and Xpress-MPB by

Dash Optimization Inc. [108]. In later chapters, we will apply branch and bound techniques to

several problems in order to find the optimal solutions and verify the effectiveness of heuristics.

2.7.2 Heuristics

Despite the guarantee of eventually reaching the optimal solution, B&B methods are

inefficient on large problems. Therefore, we must look for efficient ways of producing high

quality solutions. Heuristics, or suboptimal algorithms provide this outlet. The term heuristic is

derived from the Greek word heuriskein (evptowew),, meaning "to find or discover". Heuristics

are approximation algorithms and are the only alternative to finding "good" feasible solutions

when problems are too difficult to apply branch and bound methods. The study of heuristics is

vast, and has let to the creation of algorithmic methods which are capable of producing excellent

solutions in seconds, for problems in which a B&B or other optimal algorithm would require

years to solve.

In the following paragraphs, we provide a brief introduction to several heuristics which we

will analyze in later chapters of this dissertation. We begin with a the simplest type of heuristic

known as the greedy algorithm .

Greedy Heuristics and Local Search. A greedy algorithm is a local search metaheuristic

which gets its name from the myopic way in which it creates candidate solutions [123]. At each

step, the greedy method makes whatever choice seems best at that particular moment in time.

Once a decision is made, it is permanent and cannot be later changed. Therefore, one must ensure

that a candidate element is feasible before adding it to the incumbent solution [105].

An example of a greedy algorithm is as follows. Suppose a cashier owes a customer

$0.42 cents. The cashier can use the greedy method to determine the minimum number of

coins required for this transaction. Pseudo-code for this method is provided in Figure 2-3. The

algorithm takes as input n, the amount of change due, in this case, a = 80.42. To begin with,

one quarter is selected bringing the balance to $0.17. Next, one dime is chosen and the remainder

is $0.07. By selecting one nickel and two pennies, the problem is solved. We see that greed

is manifested in this example as the algorithm selects the highest valued coins first. For this

problem, the greedy algorithm computes the optimal solution from the 31 unique combinations of

coins which add up to $0.42 cents.

procedure GreedyChangeMaker(u)
S C <- {, 5, 10, 25}

3Sum <- 0 /s sum of coins in CI,* -,~ i~ */
4 while Sum / n do
5 x --maxcE C : Sum + c n}
6 if }9 x then
7 return: NO SOLUTION
8 else

to sum <- sum + x
11 end if
L2 end while
13return Ch. .i~~e
end procedure GreedyChangeMaker

Figure 2-3: Pseudo-code for a greedy algorithm which makes change using the minimum number
of coins.

Other problems for which the greedy method finds the optimal solution include the

MINIMUM SPANNING TREE problem where Kruskal's algorithm [122] finds a minimum

weight spanning tree of a given graph [6]. Despite the performance of the greedy algorithm

on the above example, greedy methods almost always fall short of the optimal solution when

applied to HiP-complete problems. This is because greedy methods select a local optimum from

the neighborhood of the current solution at each step with the hope that in the end, the global

optimum is found. However as we learned earlier in the chapter, this isn't necessarily the case.

Other local search heuristics involve simple examinations of neighborhoods in the quest

for a "good" solution [51]. The method moves from one solution to the next in the feasible

region, until the current solution cannot be improved by selecting an alternate solution in its

neighborhood. The specific neighborhood structure depends upon the problem at hand, and

as mentioned earlier, a clever choice of neighborhood can greatly improve the efficacy of the

heuristic. Popular local search methods include the 2-exchange (or, 2-opt) method [173], hill

climbing procedures, the method of conjugated gradients [91, 92], and steepest ascent/descent

methods. We will see several examples of local search algorithm in the later chapters. For

procedure GRASP(Maxlter, RandomSeed)
S f <-
2 X*t <--
3for i = 1 to Maxlter do
4 X <- ConstructionSolution (G, g, X, a~)
5 X <- LocalSearch(X, NV(X))
6 if fX) > fX*) then
7 X <-X
8 f f (X)
to end
11 return X*
end procedure GRASP

Figure 2-4: GRASP for maximization

detailed implementation specifications, one should consult textbook on local search, such as the

work of [2]. For an annotated bibliography of local search, the reader is also referred to [3].

Greedy Randomized Adaptive Search Procedure (GRASP).

GRASP [69] is a multi-start metaheuristic that has been used with great success to

provide solutions for several difficult combinatorial optimization problems [72], including


ASSIGNMENT [128, 140]. For an annotated bibliography of GRASP, the reader should reference

the paper by Festa and Resende [72].

GRASP is a two-phase procedure which generates solutions through the controlled use of

random sampling, greedy selection, and local search. For a given problem II, let F be the set of

feasible solutions for II. Each solution X E F is composed of k discrete components al, .., ak-

GRASP constructs a sequence {X}4 of solutions for II, such that each Xi E F. The algorithm

returns the best solution found after all iterations. The GRASP procedure can be described as in

the algorithm presented in Figure 2-4. The construction phase receives as parameters an instance

of the problem G, a ranking function g : A(X) H R (where A(X) is the domain of feasible

components al, .., ak, for a partial solution X), and a parameter 0 < a~ < 1. The construction

phase begins with an empty partial solution X. Assuming that |A(X)| = k, the algorithm creates

a list of the best ranked a~k components in A(X), and returns a uniformly chosen element x from

this list. The current partial solution is augmented to include x, and the procedure is repeated

until the solution is feasible, i.e. until X E F.

The intensification phase consists of the implementation of a hill-climbing procedure. Given

a solution X E F, let NV(X) be the set of solutions that can found from X by changing one of

the components a s X. Recall that NV(X) is called the neighborhood of X. The improvement

algorithm consists of finding, at each step, the element X* such that

X* := argmax f (X'),

where f : F R is the objective function of the problem. At the end of each step we make

X* <- X if f (X) > f (X*). The algorithm will eventually achieve a local optimum, in which

case the solution X* is such that f (X*) > f (X') for all X' E NV(X*). X* is returned as the

best solution from the iteration and the best solution from all iterations is returned as the overall

GRASP solution.

Simulated Annealing. In statistical mechanics, the physical process of annealing is used to

relax a system to the state of minimal energy. This is done by heating the solid until it melts and

then cooling it slowly so that at each temperature the particles randomly arrange themselves until

reaching thermal equilibrium.

In [116], Kirkpatrick et al. introduced a method for combinatorial problems known as

simulated annealing. Based on the theory of the physical process, simulated annealing was

shown to asymptotically converge to the global optimum after performing a number of so-called

transitions at decreasing temperatures.

Pseudo-code for a generic simulated annealing algorithm is presented in Figure 2-5. The

algorithm takes as input the initial temperature T and a reduction factor r E (0, 1). Simulated

annealing essentially chooses a neighbor at random to replace the incumbent solution. If the

chosen neighbor is a better solution then it is accepted with probability 1. However, in order to

escape and evade local optima, if the chosen neighbor is worse than the incumbent, then it is

procedure S imul ated~nne al ing (T, r)
S f <-
2 X*t <--
3 X <- randomSolution()
4 while T / 0 do
5 for i = 1 to Maxlter do
6 X' <--randomNeighbor (X)
7 if fX') > fX) then
8 X <- X'
to X <- X' with probability e T
11 end if
12 T <- rT
L3 if f (X) > f then
14 X* <-X
L5 f f (X)
16 end if
17 end for
I8 end while
19 return X*
end procedure SimulatedAnnealing

Figure 2-5: Generic simulated annealing maximization algorithm.

accepted with some positive probability which is a decreasing function of the temperature [1].

Thus the cooling schedule, or the method in which the temperature decreases is an important

part of the heuristic. It has been shown that a logarithmically slow cooling schedule guarantees

that the algorithm will converge to the global optimum in exponential time [24]. Therefore in

practice, faster cooling schedules are often used. Another method closely resembling simulated

annealing is the method of mean field annealing.

Mean field annealing (MFA), is a heuristic which mimics the idea of mean field

approximation from statistical physics [150]. In MFA, the stochastic process in simulated

annealing is replaced by a set of deterministic equations. Though MFA does not guarantee

convergence to a global optimal solution, it can provide an excellent approximation to an optimal

solution and is much less expensive computationally.

Genetic Algorithms.

procedure GeneticAlgorithm
SGenerate population PI
2 Evaluate population PI
3 while terminating condition not met do
4 Select individuals from PI and copy to Pk~+1
SCrossover individuals from PI and put in Pk~+1
6 Mutate individuals from Pk, and put in Pk~+1
7 Evaluate population PI+1

to end while
11 return best individual in Pk,
end procedure GeneticAlgorithm

Figure 2-6: Pseudo-code for generic genetic algorithm.

Genetic algorithms receive their name from an explanation of the way they behave. It comes as

no surprise, they are based on Darwin's Theory of Natural Selection [56]. Genetic algorithms

store a set of solutions, or a population, and the population evolves by replacing these solutions

with better ones based on certain fitness criterion represented by the obj ective function value.

In successive iterations, or generations, the population evolves by reproduction, crossover,

and mutation. Reproduction is the probabilistic selection of the next generations elements

determined by their fitness level. Crossover is the combination of two current solutions,

called parents which produces one or more other solutions, referred to as o~ff~spring. Finally,

mutation is the random modification of the offspring. Mutation is performed as an escape

mechanism to avoid getting trapped at a local optimum [86]. In successive generations, only

those solutions having the best fitness are carried to the next generation in a process which

mimics the fundamental principle of natural selection, survival of the fittest [56]. Figure 2-6

provides pseudo-code for a standard genetic algorithm. Genetic algorithms were introduced in

1977 by Holland [102], and were greatly invigorated by the work of Goldberg in [86].

2.8 Concluding Remarks

In this chapter we have provided a brief history and introduction to global optimization. The

chapter is not intended to be all inclusive; instead, the purpose of its inclusion is as follows. First,

we have provided the fundamental results and underlying theory that we will use throughout

this dissertation. This includes the theory of computational complexity, and a short overview of

the most common solution techniques we will encounter and apply to several problems as we

progress. Secondly, we have provided several definitions, lemmata, and theorems that we will

reference in the chapters to come. The intent is to have a concise location to which the reader can

refer. Also, presenting the maj or theorems here will prevent redundancy as we will not re-state

the theorems in each chapter they are applied. We will now move on and begin the examination

of several combinatorial problems that occur in military telecommunication networks. We

conclude this chapter with a list of references on theory, algorithms, and applications of global

and combinatorial optimization.

Excellent references on global and combinatorial optimization include the work of Du and

Pardalos [63, 64, 65], Floudas and Pardalos [74, 76], Horst and Pardalos [106], Horst, Pardalos,

and Thoai [107], Pardalos [146], Pardalos and Resende [147], Pardalos and Rosen [148], and

Wolsey [173] to name a few. Perhaps the most inclusive one-stop reference is the monumental

work of Floudas and Pardalos in the six volume Encyclopedia of Optimization [75].

The list of algorithms is also not intended to be exhaustive. Other exact algorithms include

dynamic programming [16] and outer approximation methods [107]. Other effective heuristics

include tabu search [81, 82, 83], scatter search [80], hybrid heuristics which combine elements

of several methods [22, 71, 159], and algorithms designed for the specific problem which exploit

the combinatorial structure of the problem [25, 30, 43, 139]. Other algorithmic reference books

include Ahuja et al. [6], Floudas and Pardalos [73], Goldberg [86], Minieka [130], Osman [142],

and Osman et al. [143].


3.1 Introduction

In this chapter, we study two variants of the CRITICAL NODE PROBLEM. In general, the

obj ective of the CRITICAL NODE PROBLEM (CNP) is to find a set of k nodes in a graph whose

deletion results in the maximum network fragmentation. By this we mean, maximize the number

of components in the k-vertex deleted subgraph. Studies carried out in this line include those

by Bavelas [14] and Freeman [78] which emphasize node centrality and prestige, both of which

are usually functions of a nodes degree. However, they lacked applications to problems which

emphasized network fragmentation and connectivity.

We can apply the CNP to the problem of jamming wired telecommunication networks by

identifying the critical nodes and suppressing the communication on these nodes. This will

result in the maximum number of disconnected components which are unable to communicate

with each other. The CNP can also be applied to the study of covert terrorist networks, where a

certain number of individuals have to be identified whose deletion would result in the maximum

breakdown of communication between individuals in the network [118]. Likewise in order to

stop the spreading of a virus over a telecommunication network, one can identify the critical

nodes of the graph and take them offline.

The CNP also finds applications in network immunization [36, 176] where mass vaccination

is an expensive process and only a specific number of people, modeled as nodes of a graph,

can be vaccinated. The immunized nodes cannot propagate the virus and the goal is to identify

the individuals to be vaccinated in order to reduce the overall transmissibility of the virus.

There are several vaccination strategies in the literature [36, 176] offering control of epidemic

outbreaks; however, none of the proposed are optimal strategies. The vaccination strategies

suggested emphasize the centrality of nodes as a maj or factor rather than critical nodes whose

deletion will maximize disconnectivity of the graph. Deletion of central nodes may not guarantee

a fragmentation of the network or even disconnectivity, in which case disease transmission

cannot be prevented. Of course, owing to its dynamic stature, the relationships between people,

represented by edges in the social network are transient and there is a constant rewiring between

nodes, and alternate relationships could be established in the future. The proposed critical

node technique helps in a maximum prevention of disease transmission over an instance of the

dynamic network.

Before proceeding, we mention one final area in which the CRITICAL NODE PROBLEM finds

several applications, and that is in the field of transportation engineering [66]. Two particular

examples are as follows. In general, for transportation networks, it is important to identify critical

nodes in order to ensure they operate reliably for transporting people and goods throughout the

network. Further, in planning for emergency evacuations, identifying the critical nodes of the

transportation network is crucial. The reason is two-fold. First, knowledge of the critical nodes

will help in planning the allocation of resources during the evacuation. Secondly, in the aftermath

of a disaster they will help in re-establishing critical traffic routes.

Borgatti [21] has studied a similar problem, focusing on node detection resulting in

maximum network disconnectivity. Other studies in the area of node detection such as centrality

[14, 78] focus on the prominence and reachability to and from the central nodes. However, little

emphasis is placed on the importance of their role in the network connectivity and diameter.

Perhaps one reason for this is because all of the aforementioned references relied on simulation to

conduct their studies. Although the simulations have been successful, a mathematical formulation

is essential for providing insight and helping to reveal some of the fundamental properties of

the problem [138]. In the next section, we present a mathematical model based on integer linear

programming which provides optimal solutions for the CRITICAL NODE PROBLEM.

We organize this chapter by first formally defining the problem and discussing its

computational complexity. Next, we provide an integer programming (IP) formulation for

the corresponding optimization problem. In Section 3.3 we introduce a heuristic to quickly

provide solutions to large-scale instances of the problem. We present a computational study in

Section 3.4, in which we compare the performance of the heuristic against the optimal solutions

which were determined using a commercial software package. Some concluding remarks are

given in Section 3.5.
3.2 Problem Formulations

Denote a graph G = (V, E) as a pair consisting of a set of vertices V, and a set of edges E.

All graphs in this chapter are assumed to be undirected and unweighted. For a subset W c V, let

G(W) denote the subgraph induced by W on G. A set of vertices I CV is called an independent

or stable set if for every i, j E I, (i, j) 5' E. That is, the graph G(I) induced by I is edgeless. An

independent set is maximal if it is not a subset of any larger independent set (i. e., it is maximal by

inclusion), and maximum if there are no larger independent sets in the graph.

3.2.1 Critical Node Problem

The formal definition of the problem is given by:


INPUT: An undirected graph G = (V, E) and an integer k.

OUTPUT: A1 = a~rg min CiljE(V\A) 1ij (G(V \ A)) : |A| < k, where

1, if i and j are in the same component of G(V \ A)
Uij 10, otherwise.

The objective is to find a subset A cV of nodes such that |A| < k, whose deletion results

in the minimum value of C uij in the edge induced sub graph G(V \ A). This obj ective function

results in a minimum cohesion in the network, while also ensuring a minimum difference in the

sizes of the components. An illustration is best suited to explain the choice of obj ective function.

Consider an arbitrary unweighted graph with 150 nodes. According to our obj ective, it is more

preferable to have a partition with 3 components with each 50 nodes as opposed to a partition

with 5 components with one having 146 nodes and the rest of them having a single node.

This problem is similar to MINIMUM k-VERTEX SHARING [133], where the obj ective is

to minimize the number of nodes deleted to achieve a k-way partition. Here we are considering

the complementary problem, where we know the number of vertices to be deleted and we try to

maximize the number of components formed and implicitly limit the sizes of the components.

Borgatti [21] has given a comprehensive illustration to facilitate the understanding of the

obj ective function and its non-triviality.

We now prove that the recognition version of the CNP is HiP-complete. Consider the

following decision problem for the CNP:


INPUT: An undirected graph G = (V, E) and an integer k.

QUESTION: Does there exist a zero cost K-way partition of G by deleting k nodes or less?

Theorem 5. Thze K-CRITICAL NODE PROBLEM is HiP-complete.

Proof: To show this, we must prove that (1) K-CNP E NiP; (2) Some NiP-complete problem

reduces to K-CNP in polynomial time.

(1) K-CNP E NiP since given any graph G = (V, E), we can verify the validity of G in

polynomial time. More specifically, by deleting any set of at most k nodes, we determine

if the there is a zero-cost K-way partition of G in O(| E| + |V|) time using a depth-first

search [6].

(2) To complete the proof, we show a reduction from the K-INDEPENDENT SET PROBLEM

(K-ISP) [24], which is well-known to be HiP-complete [79]. Recall that the obj ective of

the K-ISP is to determine if G contains an independent set containing at least K nodes.

Let G = (V, E) be a graph in which we seek an independent set. There are no necessary

transformations required for the graph in which we are solving the corresponding K-CNP.

We will show that a 'yes' instance of the K-ISP corresponds to a 'yes' instance of the

K-CNP on G. In particular, G has an independent set of size K if and only if the K-CNP

has a zero cost solution where k < |V| K. Suppose G contains an independent set I

where |7| = K. Notice that the objective of the K-CNP will be 0 as the subgraph induced

by deleting the nodes in V \ I is edgeless. Therefore, a 'yes' instance of the K-ISP implies

a 'yes' instance for the K-CNP with k~ = |V| K.

To prove the converse, observe that the cost of any K-CNP is at least 0. Thus, a 'yes'

instance of the K-CNP would imply that once the k critical nodes are removed, the resulting

subgraph consists of K components whose obj ective function is 0. This implies that the

induced subgraph is edgeless, i.e. each of the K components consists of a single node.

Hence, the K remaining nodes form an independent set of G, resulting in a 'yes' instance

for the K-INDEPENDENT SET PROBLEM. Thus the proof is complete.

When studying combinatorial problems, integer programming models are usually quite

helpful for providing some of the formal properties of the problem [13 8]. With this in mind we

now develop a linear integer programming formulation for the CNP.

To begin with, define the surjection it : V xVB { 0, 1} as above. Further, we introduce a

surjection t' : VB {0(, 1} defined by

1, if node i is deleted in the optimal solution,

0, otherwise. 31

Then the CRITICAL NODE PROBLEM admits the following integer programming formulation

(CNP-1) Minimize C L/(3-2)


ujk Zi < 1j> V (i, j,) E V, (3-4)

i .,Ilki< 1 V (, j k) V,(3-5)

It 't + lki< 1,V (, jk) V,(3-6)

<' kl (3-7)

Its {0,1}, i, e V,(3-8)

E 0, 1, Vi V.(3-9)

Theorem 6. CNP-1 is a correct formulation for the CRITICAL NODE PROBLEM.

Proof: First, we note that the obj ective is to find the set of k nodes whose removal results in a

graph which has the maximum number of disconnected components. This is accomplished by

the obj ective function. Notice that the first set of constraints in (3-3) implies that if nodes i and

j are in different components and if there is an edge between them, then one of them must be

deleted. Furthermore, constraints (3-4)-(3-6) together imply that for all triplets of nodes i, j, k,

that if i and j are in same component and j and k are in same component, then k and i must be in

the same component. Constraint (3-7) ensures that the total number of deleted nodes is less than

or equal to k. Finally, (3-8) and (3-9) define the proper domains for the variables used. Thus,

a solution to the integer programming formulation CNP-1 characterizes a feasible solution to

the CNP. On the other hand, it is clear that a feasible solution to the CNP will define at least one

feasible solution to CNP-1. Therefore, CNP-1 is a correct formulation for the CNP. O

Notice that the conditions which satisfy the circular constraints (3-4), (3-5), and (3-6) in

CNP-1 can be satisfied by the single constraint uij + "/ ., + uki / 2, V (i, j, k) E V Thus we have

an equivalent, more compact integer program given as

(CNP-2) Minimize u p, (3-10)


ui + + vj > 1, V (i, j) E E, (3-11)

ui ujk + uki / 2, V (i, j, k) E V, (3-12)

<' kl (3-13)

u E 0, 1, Vi, j V,(3-14)

E 0, 1, Vi V,(3-15)

where ui,j and I are as defined above.

Notice that if the obj ective function had only the number of components, then an

approximation for the MAXIMUM K-CUT PROBLEM [79, 112] could be employed by modifying

the cost function of the Gomory-Hu tree [89]. An even simpler approach would be to identify

the cut vertices in the graph, if any exist. However, the obj ective function also involves the sizes

of the components formed, which makes the problem harder and subsequently implies that the

methods suggested above are not suitable for our problem.

Recall that CijE m is a measure of the total disconnectivity of the graph. If we observe

carefully, the obj ective function could be rewritten as

E 2 (3-16)

where S is set of all components and as is the size of the ith component, which can be easily

identified by fast algorithms like breadth or depth first search algorithms in O(|V| + |E|) time

[48]. We now provide an intuitive explanation for the choice of our obj ective function. For a fixed

number of components the variance in the sizes of the components will be the sum of the squares

of deviation of sizes of the components from the mean size of a component. However notice that

the mean size of any component is constant because the sum of the sizes of the components is

the constant, |V| k. Thus minimizing the variance of the size of the components reduces to

minimizing the sum of squares of the sizes of the components, which is our obj ective function.

Also, when the sizes of the components are equal the obj ective function is the minimum when

the number of components is the maximum. We will use this obj ective function in the following

section to implement a heuristic for identifying critical nodes.

3.2.2 Cardinality Constrained Problem

We now provide the formulation for a slightly modified version of the CNP based on

constraining the connectivity index of the nodes in the graph. Given a graph G = (V, E),

the connectivity index of a node is defined as the number of nodes reachable from that vertex.

Examples are provided in Figure 3-1. To constrain the network connectivity in optimization

models, we can impose constraints on the connectivity indices.

This leads to a cardinality constrained version of the CNP which we aptly refer to as



Figure 3-1: Connectivity Index of nodes A,B,C,D is 3. Connectivity Index of E,F,G is 2.
Connectivity Index of H is 0.

obj ective is to detect a set of nodes A cV such that the connectivity indices of the nodes in

the vertex deleted subgraph G(V \ 4) is less than some threshold value, say L. Using the same

definition of the variables as in the previous subsection, we can formulate the CC-CNP as the

following integer linear programming problem.

(CC-CNP-1) Minimize Cr(3-1 7)


uij+ i+ v > 1, V (i, j) E E, (3-18)

u jk + Z~ ki yd 2, V (i, j, k) E V, (3-19)

< ~ L, (3-20)

is E{0, }, Vi, j V,(3-21)

E 0, 1, Vi V,(3-22)

where L is the maximum allowable connectivity index for any node in V.

Theorem 7. CC-CNP1 is a correct formulation for the CARDINALITY CONSTRAINED CRITICAL


Proof: This proof follows in much the same way as Theorem 6. First, we see that the obj ective

function given clearly minimizes the number of nodes deleted. Constraints (3-18) and (3-19)

follow exactly as in the CNP formulation. The only difference is now we must constrain

procedure CriticalNode(G, k~)
S MIS <- MaximallndepSet (G)
2 while (|MIS| / |V| k) do
3 i -- arMri(, ies : SE G(MIS U {i}), ie V I\MIS}
4 MIS <- MIS U {i}
5 end while
6 return V \ MIS /s set of k nodes to delete s/
end procedure CriticalNode

Figure 3-2: Heuristic for detecting critical nodes.

the connectivity index of each node. This is accomplished by constraint (3-20). Finally

constraints (3-21) and (3-22) define the domains of the decision variables, and we have the

proof. O

3.3 Heuristics for Critical Node Problems

3.3.1 CNP Heuristic

Pseudo-code for the proposed heuristic is provided in Figure 3-2. To begin with, the

algorithm finds a maximal independent set (MIS). Then in the loop from lines 2-5, the heuristic

greedily selects the node i E V not currently in MIS which returns the minimum objective

function for the graph G(MIS U {i}). The set MIS is augmented to include node i, and the

process repeats until |MIS|I = |V| k. The method terminates and the set of critical nodes to be

deleted is given as those nodes j E V such that j e V \ MIS.

The intuition behind using an independent set is that the subgraph induced by this set is

empty. Stated otherwise, the deletion of nodes that are not in the independent set from the graph

will result in an empty subgraph. Notice that this will provide the optimal solution for an instance

of the CNP if |MIS| > |V| k. However, if the size of MIS is less than |V| k, we simply

keep adding nodes which provide the best obj ective value to the set until it reaches the desired

size. In the following lemma, we establish a relationship between the CNP and the MAXIMUM

INDEPENDENT SET problem, which also provides a bound on the optimal solution for an instance

of the CNP.

Lemma 5. Given a graph G = (V, E), the cardinality of the maximum independent set of G,

denoted a~ ( G) provides an upper bound on the number of components produced in the optimal

solution of the corresponding CRITICAL NODE PROBLEM for any value of k E Z.

Proof: Obviously, removing the critical nodes determined by the optimal solution for any

instance of the CNP results in a set of disconnected components of G. One node from each

of these components forms an independent set. Hence a~(G) should be at least as large as the

number of components formed in the optimal solution to the CNP. Furthermore, the components

formed in the subgraph induced by the maximum independent set are of size one, and hence

result in the optimal solution for the CNP instance if a~(G) > |V| k, i.e. if the deletion of some

k nodes results in an empty graph. Thus, we have the lemma. O

We note that this bound is not particularly useful in practice since the MAXIMUM

INDEPENDENT SET problem is HiP-hard in general [24, 79]. However, a maximal independent

set can be computed in polynomial time. This motivates our decision to use maximal instead

of maximum independent sets in the heuristic. Subsequently the heuristic is computationally

efficient, with the complexity given in the following theorem.

Theorem 8. T2e proposed algorithm has complexity O(k2 + |Vlk)).

Proof: To begin with, the while loop from lines 2-5 will iterate at most O(|V| k) times. In each

iteration, the number of search operations decreases from |V| 1 to |V| (|V| k) = k. Note

that we are performing the search of a sparse graph, which is initially empty. Hence the total

complexity will be

o(|V| -1+|IV|- 2 + -+|IV| -|IV|+k) = O C iIVV- i = O(k2 +|IVk).
i= 1 i= 1

Thus the proof is complete. O

The proposed algorithm finds a feasible solution to the CRITICAL NODE PROBLEM;

however, the solution is not guaranteed to be globally or locally optimal. Therefore, we can

enhance the heuristic with the application of local search routine as follows. Consider the

procedure LocalSearch(V \ MIS)
S X* <- MIS
2 local _improvement <- .TRUE.
3 while local _improvement do
4 local _improvement <- .FALSE.
5 if i s MIS and j 5( MIS then
6 MIS <- MIS \ i
7 MIS <- MIS U j
8 if f (\Ii) < f (X*) then
9 ~X* <- MIS
to local_improvement <- .TRUE.
11 end if
L2 end if
13end while
14 return (V \ X*) /s set of k nodes to delete s/
end procedure LocalSearch

Figure 3-3: Local search algorithm for critical node heuristic.

pseudo-code presented in Figure 3-3. The routine receives as input the solution from the

CriticalNode heuristic and performs a 2-exchange local search. Let f : VB Z be a function

returning the obj ective function value for a given set, in the sense of (3-16) above. That is,

consider a pair of nodes i and j such that i s MIS and j 5( MIS. Then for all such pairs, we set

j E MIS and i 5 MIS and examine the change in the objective function. If it improves, then

the swap is kept; otherwise, we undo the swap and continue to the next node pair. Notice that the

loop from lines 3-13 repeats while the solution is not locally optimal. This general statement can

lead to implementation problems and it is a common practice to limit the number of local search

iterations by some user defined value, say U. The intuition is that the as U 00o, the solution

becomes optimal with respect to its local neighborhood.

Theorem 9. If the number of iterations of the local search is bounded by a constant U E R as

described above, then the complexity of the procedure is O ( |V |2U

Proof: The is clear as the while loop from lines 3-13 will iterate U times. Since each iteration

requires an examination of |V|2 COmponents, we have the proof. O

procedure CriticalNodeLS(G, k)
S X* <-
2 f (X*)t <-
3 for j = 1 to Maxlter do
4 X <- CriticalNode(G, k~)
5 X <- LocalSearch(X)
6 iff (X) < f (X*) then
7 X <-X
8 end if
to return (V \ X*) /s set of k nodes to delete s/
end procedure CriticalNodeLS

Figure 3-4: Heuristic with local search for detecting critical nodes.

Finally, we can combine the construction and local improvement algorithms into one

multi-start heuristic CriticalNodeLS as shown in Figure 3-4. This procedure produces Maxlter

local optima and the overall best solution from all iterations is returned.

Theorem 10. The CriticalNodeLS heuristic has overall complexity of O(|V|2UT (k2 + |Vlk)),

where T = Maxlter, and' U is the iteration limit on the local search.

Proof: This result follows directly from Theorem 8 and Theorem 9 above. O

3.3.2 CC-CNP Heuristic

With a subtle modification to the heuristic described above for the CNP, we can create an

effective heuristic for the CC-CNP. To do this, notice that now we are only concerned with the

connectivity indices of the nodes. Stated differently, we are only concerned with the sizes of the

components in the vertex deleted sub graph. Unlike before, there is no limit on the number of

critical nodes we choose, so long as the connectivity constraints are satisfied.

Pseudo-code for the proposed algorithm is provided in Figure 3-5. The heuristic starts

off the same as before by identifying a maximal independent set (MIS). Then, the boolean

variable OPT is set to FALSE. Finally in line 3, a variable NoAdd is initialized to 0. This

variable determines when to exit the main loop from lines 4-16. After this loop is entered, the

procedure iterates through the vertices and determines which can be added back to the graph

while still maintaining feasibility. If vertex i can be added, MIS is augmented to include i in step

procedure ConstrainedCriticalNode (G, L)
S MIS <- MaximallndepSet (G)
3 NoAdd <- 0
4 while (OPT .NOT.TRUE) do
5 for (i =1 to|IV|)do
Eiif (Is~E < L, VeS c G;(MIS U {i})) : ie V \MIS) then
7 MIS <- MIS U {i}
8 else
9 ~NoAdd <- NoAdd +1
to end if
11 if (NoAdd = |V| |MIS|) then
12 OPT <- TRUE
14 end if
15 end for
16 end while
17 return V \ MIS /s set of nodes to delete s/
end procedure ConstrainedCriticalNode


7, otherwise NoAdd is incremented. If NoAdd is ever equal to |V| |MIS|i, then no nodes can be

returned to the graph and OPT is set to TRUE. Then loop is then exited and the algorithm returns

the set of nodes to be deleted, i.e. V\MIS.

Theorem 11. The worst-ca~se complexity of the ConstrainedCriticalNode heuristic is

O(|V|2 + |V||E|).

Proof: This proof is similar to the proof of Theorem 8 above. The loop from lines 4-16 will

iterate at most O(|V|) times. Each loop requires at most O(|V| + |E|) time to verify the

if a solution will remain feasible after a node is re-included in the graph. Thus we have the

result. O

3.3.3 Genetic Algorithm for the CC-CNP

As mentioned in Subsection 2.7.2, genetic algorithms (GAs) mimic the biological process of

evolution. In this subsection, we describe the implementation of a GA for the CC-CNP. Recall the

general structure of a GA as outlined in Figure 3-6. When designing a genetic algorithm for an

procedure GeneticAlgorithm
SGenerate population PI
2 Evaluate population PI
3 while terminating condition not met do
4 Select individuals from PI and copy to PIt+
5 Crossover individuals from PI and put in PI 1
6 Mutate individuals from PI and put in PI 1
7 Evaluate population PI 1

9 P,+1t Q
to end while
11 return best individual in PI
end procedure GeneticAlgorithm

Figure 3-6: Pseudo-code for a generic genetic algorithm.

optimization problem, one must provide a means to encode the population, define the crossover

operator, and define the mutation operator which allows for random changes in offspring to help

prevent the algorithm from converging prematurely [10].

For our implementation, we use binary vectors as an encoding scheme for individuals within

the population of solutions. When the population is generated, (Figure 3-6, line 1), a random

deviate from a distribution which is uniform onto (0, 1) E R is generated for each node. If the

deviate exceeds some specified value, the corresponding allele is assigned value 1, indicating

this node should be deleted. Otherwise, the allele is given a 0, implying it is not deleted. In order

to evaluate the fitness of the population, per line 2, we must determine whether each individual

solution is feasible or not. Determining feasibility is a relatively straightforward task and can

accomplished in O(|V| + |E|) using a depth-first search [6].

In order to evolve the population over successive generations, we use a reproduction scheme

in which the parents chosen to produce the offspring are selected using the binary tournament

method [131, 172]. Using this method, two chromosomes are chosen at random from the

population and the one having the best fitness, i.e. the lowest obj ective function value, is kept

as a parent. The process is then repeated to select the second parent. The two parents are then

combined using a crossover operator to produce an offspring [94].

Coin Toss T H H T H
MOM 0.56 0.81 0.22 0.7 0.86;
Offspring 0.81 0.22 0.86;

Figure 3-7: Example of the crossover operation. In this case, CrossProb = 0.65.

To breed new solutions, we implement a strategy known as parameterized uniform crossover

[167]. This method works as follows. After the selection of the parents, refer to the parent

having the best fitness as MOM. For each of the nodes alleless), a biased coin is tossed. If the

result is heads, then the allele from the MOM chromosome is chosen. Otherwise, the allele from

the least fit parent, call it DAD, is selected. The probability that the coin lands on heads is known

as CrossProb, and is determined empirically. Figure 3-7 provides an example of a potential

crossover when the number of nodes is 5 and CrossProb = 0.65 [10].

After the child is produced, the mutation operator is applied. Mutation is a randomizing

agent which helps prevent the GA from converging prematurely and escape to local optima.

This process works by flipping a biased coin for each allele of the chromosome. The probability

of the coin landing heads, known as the mutation rate (MutRate) is typically a very small user

defined value. If the result is heads, then the value of the corresponding allele is reversed. For our

implementation, MutRate = 0.03.

After the crossover and mutation operators create the new offspring, it replaces a current

member of the population using the so-called steady-state model [37, 94, 13 1]. Using this

methodology, the child replaces the least fit member of the population, provided that a clone

of the child is not an existing member in the population. This method ensures that the worst

element of the population is monotonically improving in every generation. In the subsequent

iteration, the child becomes eligible to be a parent and the process repeats. Though the GA does

converge in probability to the optimal solution, it is common to stop the procedure after some

"terminating condition" (Figure 3-6, line 3) is satisfied. This condition could be one of several

things including, a maximum running time, a target obj ective value, or a limit on the number of

Table 3-1: Results of IP model and heuristic on terrorist network data.
Instance IP Model Heuristic Heuristic + LS
Nodes Obj ective Execution Objective Execution Objective Execution
Deleted (k) Value Time (s) Value Time (s) Value Time (s)
20 20 12.69 22 0.08 20 0.01
15 61 277.77 66 0.03 61 0.01
10 169 3337.06 190 0.06 169 0.02
9 214 2792.33 229 0.15 214 0.02
8 282 15111.94 309 0.04 282 0.01
7 327 10792.08 329 0.09 327 0.01

generations. For our implementation, we use the latter option and the best solution after MaxGen

generations is returned.

3.4 Computational Results

All of the proposed heuristics were implemented in the C++ programming language and

complied using GNU g++ version 3.4.4, using optimization flags -02. It was tested on a PC

equipped with a 1700MHz IntelB PentiumB M processor and 1.0 gigabytes of RAM operating

under the MicrosoftB WindowsB XP Professional environment.

3.4.1 CNP Results

We begin with the numerical results of the combinatorial algorithm for the CRITICAL NODE

PROBLEM. We tested the IP model and the aforementioned heuristic on the terrorist network

from Krebs [1 18] as well as on a set of randomly generated scale-free [13] graphs ranging in size

from 75 to 150 nodes with various densities. The graphs were generated with version 1.4 of the

publicly available Barabasi graph generator by Dreier [62]. For each instance tested, we report

solutions for 3 values of k, the number of nodes to be deleted.

As a basis for comparison, we have implemented the integer programming model for the

CRITICAL NODE PROBLEM using the CPLEXTM Optimization suite from ILOG [50]. CPLEX

contains an implementation of the simplex method [98], and uses a branch and bound algorithm

[173] together with advanced cutting-plane techniques [107, 139].

We begin by providing the results from the terrorist network [118]. The graph, which is

shown in Figure 3-8 has 62 nodes and 153 edges. Notice that node 38 is the central node with

Figure 3-8: Terrorist network compiled by Krebs.

degree 22. We applied the IP formulation and the heuristic to this network with 6 values of k.

The results are provided in Table 3-1. Notice that for all values of k, the heuristic computed the

optimal solution requiring on average 0.013 seconds of computation time. The average time to

compute the optimal solution using CPLEX was 5387.31 seconds. Clearly even for this relatively

small network, the heuristic is the method of choice. Figure 3-9 shows the resulting graph of the

terrorist network according to the optimal solution to the CNP for the instance of k = 20.

In order to determine the scalability and robustness, the proposed heuristic was tested on a

set of randomly generated scale-free graphs. Table 3-2 presents the results of the heuristic and the

optimal solver when applied to the random instances. For each instance, we report the number

of nodes and arcs, the value of k being considered, the optimal solution and computation time

required by CPLEX, and finally the heuristic solution and the corresponding computation time.

For each graph, we report solutions for 3 different values of k.

f48 40' 46
4( 426 25


34~2 i 1

g 17

Figure 3-9: Optimal solution when k = 20.

Notice that for all instances tested, our method was able to compute the optimal solution.

Furthermore, the required time to compute the optimal solution was less than one second for all

but one instance, averaging only 0.33 seconds for all 27 instances. On the other hand, CPLEX

required 289.44 seconds on average to compute the optimal solution, requiring over 5000 seconds

in the worst case. Our computational experiments indicate that the proposed heuristic is able to

efficiently provide excellent solutions for large-scale instances of the CNP.

3.4.2 CC-CNP Results

We continue with the results of the two algorithms developed for the CC-CNP, namely

the combinatorial algorithm and the genetic algorithm. As above, we tested the IP model and

both heuristics on the terrorist network [1 18] and a set of randomly generated graphs. For each

instance tested, we report solutions for 3 values of L, the connectivity index threshold. Finally,

we have implemented the integer programming model for the CC-CNP using CPLEXTM

Table 3-2: Results of IP model and heuristic on randomly generated scale free graphs.
Instance IP Model Heuristic Heuristic + LS
Nodes Arcs Deleted Obj Comp Obj Comp Obj Comp
Nodes (k) Value Time (s) Value Time (s) Value Time (s)
75 140 20 36 66.7 92 0.12 36 0.03
75 140 25 18 33.28 39 0.28 18 0.03
75 140 30 7 4.23 18 0.02 7 0.04
75 210 25 26 93.71 78 0.1 26 0.04
75 210 30 8 3.57 31 0.05 8 0.05
75 210 35 2 4.36 16 0.18 2 0.04
75 280 33 26 749.19 54 0.00 26 0.04
75 280 35 20 164.34 38 0.09 20 0.06
75 280 37 13 83.98 24 0.39 13 0.11
100 194 25 44 151.14 142 0.731 44 0.09
100 194 30 20 59.66 72 0.56 20 0.11
100 194 35 10 8.51 33 0.66 10 0.12
100 285 40 23 136.47 48 1.151 23 0.11
100 285 42 17 263.82 38 0.4 17 0.17
100 285 45 11 16.78 29 0.53 11 0.23
100 380 45 22 128.13 58 0.58 22 0.15
100 380 47 16 243.07 42 1.191 16 0.16
100 380 50 10 228.72 23 0.31 10 0.11
125 240 33 62 5047.511 97 0.721 62 0.30
125 240 40 29 118.92 49 1.5632 29 0.24
125 240 45 16 17.09 32 0.14 16 0.39
150 290 40 40 41.6 125 1.832 40 0.47
150 290 50 12 26.29 64 2.773 12 0.831
150 290 60 1 24.92 35 1.091 1 0.851
150 435 61 19 29.55 53 2.313 19 0.741
150 435 65 13 31.45 37 0.991 13 1.952
150 435 67 11 37.91 31 0.52 11 0.801

Table 3-3 presents computational results of the IP model and heuristic solutions when tested

on the terrorist network data. Notice that for all 5 values of L tested, the genetic algorithm and

the combinatorial algorithm with local search (ComAlg + LS) computed optimal solutions.

Figure 3-10 shows the optimal solution for the case when L = 4.

We now consider the performance of the algorithms when tested on the randomly generated

data sets containing up to 50 nodes taken from [9]. The results are shown in Table 3-4. For these

relatively small instances, we were able to compute the optimal solutions using CPLEX. For each

Table 3-3: Results of IP model and heuristics on terrorist network data.
Instance IP Model Genetic Alg ComAlg ComAlg + LS
Max Conn. Obj Comp Obj Comp Obj Comp Obj Comp
Index (L) Val Time (s) Val Time (s) Val Time (s) Val Time (s)
3 21 188.98 21 0.25 22 0.01 21 0.1
4 17 886.09 17 0.741 19 0.01 17 0.45
5 15 30051.09 15 0.871 20 0.18 25 1.331
8 13 0.39 14 0.05 13 0.07
10 1- 11 0.741 12 0.07 11 0.05





*** **1 **

Figure 3-10: Optimal solution when L

instance, we provide solutions for 3 values of L, the maximum connectivity index. Notice that for

these problems, the genetic algorithm computed optimal solutions for each instance tested in a

fraction of the time required by CPLEX. The combinatorial heuristic found optimal solutions for

all but 3 cases requiring approximately half of the time of the GA.

Table 3-5 presents the solutions for the random instances from 75 to 150 nodes [9, 11].

Again, in order to demonstrate the robustness of the heuristics, we provide solutions for 3

the maximum network disconnectivity. In general, the problem of detecting critical nodes has

a wide variety of applications from jamming communication networks and other anti-terrorism

applications, to epidemiology and transportation science [9, 11].

In particular we examined two problems, namely the CRITICAL NODE PROBLEM (CNP) as

well as the CARDINALITY CONSTRAINED CNP (CC-CNP). Given a graph and an integer k, the

obj ective of the CNP is to detect a set of k critical nodes whose deletion results in the maximum

number of disconnected components whose cardinalities have the minimum variance. The

definition of the CC-CNP is slightly different in that instead of given k E Z, the maximum number

of nodes to delete, we are given some value L EZ which represents the maximum connectivity

index a node may have. The obj ective in this case is to delete the minimum number of nodes

while ensuring that the connectivity index of each node does not exceed L.

The proposed problems were modeled as integer linear programming problems. Then we

proved that the corresponding decision problems are HiP-complete. Furthermore, we proposed

a several heuristics for efficiently computing quality solutions to large-scale instances. The

heuristic proposed for the CNP was a combinatorial algorithm which exploited properties of

the graph in order to compute basic feasible solutions. The method was further intensified by

the application of a local search mechanism. By using the integer programming formulation

we were able to determine the precision of our heuristic by comparing their relative solutions

and computation times for several networks. The computational experiments indicated that the

heuristic found optimal solutions for all instances tested in a fraction of the time required by the

commercial IP solver CPLEX.

For the CC-CNP we proposed two algorithms, namely a modified version of the

combinatorial algorithm described above and a genetic algorithm [87]. Once again, the

computational experiments indicated that both methods are robust and are able to efficiently

compute approximate solutions for instances up to 150 nodes.

We also conclude with a few words on the possibility of future expansion of this work. A

heuristic exploration of cutting plane algorithms on the IP formulation would be an interesting

alternative. Other heuristic approaches worthy of investigation include hybridizing the genetic

algorithm with the addition of a local search or path-relinking enhancement procedure [85].

Finally, the local search used in the combinatorial algorithm was a simple 2-exchange method,

which was the cause of a significant slow down in computation as noted in Table 3-5. A more

sophisticated local search such as a modification of the one proposed by Resende and Werneck

[159, 160] should be a maj or focus of attention.

Furthermore, it would be interesting to study the weighted version of the problem to see

how weights added to the nodes affect the solutions. For example, it is rational to perceive

applications containing weighted networks in which the cost of deleting one node is different

from another. Also, pertaining to applications outside the scope of jamming networks, a study of

epidemic threshold variation with respect to the heuristic results will help determine the impacts

on contagion suppression in biological and social networks.


4.1 Introduction

Military strategists are constantly seeking ways to increase the effectiveness of their

force while reducing the risk of casualties. In any adversarial environment, an important

goal is always to neutralize the communication system of the enemy. In this chapter, we are

interested in j amming a wireless communication network. Specifically, we study the problem

of determining the optimal number and placement for a set of jamming devices in order to

neutralize communication on the network. This is known as the WIRELESS NETWORK JAMMING


Despite the enormous amount of research on telecommunication systems [155], the topic of

jamming communication networks has received little attention. In fact, the material that follows

in the next two chapters present the first such efforts, in so far as we can tell. We will begin

this chapter by describing and formulating the problem of jamming a wired telecommunication

network, and extend this result to the wireless domain. We will see that there is a bit more

versatility when considering the wireless version of the problem due to the wireless multicast

advantage, i.e. the ability of wireless transmitters to communicate affect nodes that are not

directly adj acent to them.

We can generalize the work of [9] to study the problem of jamming and eavesdropping

wireless communication networks. As we will see, there are several variations that can be made

depending on the overall obj ectives. This is aided by the fact that wireless j amming devices

not only affect those nodes which are directly adj acent to them; rather, they propagate energy

throughout the network to all the communication nodes as we will see in the next section.

The organization of the chapter is as follows. After a review of related work, we present

several deterministic formulations of the WNJP in Section 4.3. In particular, Subsection

4.3.1 contains several coverage formulations of the WNJP. Then in Subsection 4.3.2, we use

tools from graph theory to define the connectivity of the network and develop an alternative

formulation based on constraining the connectivity indices of the nodes, analogous to the

CC-CNP. Next, in Section 4.4 we incorporate percentile constraints to develop formulations

which are computationally more efficient and have similar solution quality. In Section 4.5, we

will present two case studies comparing the solutions and computation time for all formulations.

Finally, conclusions and future directions of research will be addressed.

4.2 Definitions and Assumptions

Before formally defining the problem statement, we will state some basic assumptions about

the j amming devices and the communication nodes being j ammed. We assume that parameters

such as the frequency range of the j amming devices are known. In addition, the j amming devices

are assumed to have omnidirectional antennas. The communication nodes are also assumed to be

outfitted with omnidirectional antennas and function as both receivers and transmitters. Given a

graph G = (V, E), we can represent the communication devices as the vertices of the graph. An

undirected edge would connect two nodes if they are within a certain communication threshold.

Given a set MZ/ = {1i, 2, ..., m} of communication nodes to be jammed, the goal is to

find a set of locations for placing j amming devices in order to suppress the functionality of the

network. The jamming effectiveness of device j is calculated using d : (V x V) H IR, where

d is a decreasing function of the distance from the j amming device to the node being j ammed.

Here we are considering radio transmitting nodes, and correspondingly, jamming devices which

emit electromagnetic waves. Thus the j amming effectiveness of a device depends on the power

of its electromagnetic emission, which is assumed to be inversely proportional to the squared

distance from the j amming device to the node being j ammed. We note that this assumption is

made without the loss of generality. The results presented in this chapter hold as long as the

function d is a smooth monotonically decreasing function. Specifically,

where A E R is a constant, and r(i, j) represents the distance between node i and j amming device

j. Without the loss of generality, we can set A = 1.

possible placement locations will likely be limited. Define the decision variable xj as

1, if a jamming device is installed at location j,

xj:10, otherwise. 43

If we redefine r(i, j) to be the distance between communication node i and j amming location j,

then we have the OPTIMAL NETWORK COVERING (ONC) formulation of the WNJP as

(ONC) Minimize cyy(4-4)

Xj E {0,1, j=12,.,, (4-6)

where Ci is defined as above. Here the obj ective is to minimize the number of jamming devices

used while achieving some minimum level of coverage at each node. The coefficients cj in (4-4)

represent the costs of installing a j amming device at location j. In a battlefield scenario, placing

a jamming device in the direct proximity of a network node may be theoretically possible;

however, such a placement might be undesirable due to security considerations. In this case, the

location considered would have a higher placement cost than would a safer location. If there are

no preferences for device locations, then without the loss of generality,

cj = 1, j=1 ,...,n

Though we have removed the non-convex covering constraints, this formulation remains

computationally difficult. Notice that oNC is formulated as a MULTIDIMENSIONAL KNAPSACK

PROBLEM which is known to be HiP-hard in general [79].

4.3.2 Connectivity Formulation

In the general WNJP, it is important that the distinction be made that the obj ective is not

simply to j am all of the nodes, but to destroy the functionality of the underlying communication

network. In this section, we use tools from graph theory to develop a method for suppressing


Figure 4-1: Connectivity Index of nodes A,B,C,D is 3. Connectivity Index of E,F,G is 2.
Connectivity Index of H is 0.

the network by jamming those nodes with several communication links and derive an alternative

formulation of the WNJP. Given a graph G = (V, E), the recall that the connectivity index of

a node is defined as the number of nodes reachable from that vertex (as shown in Figure 4-1).

To constrain the network connectivity in optimization models, we can impose constraints on the

connectivity indices instead of using covering constraints.

We can now develop a formulation for the WNJP based on the connectivity indices of the

communication graph. We assume that the set of communication nodes MZ/ = {1, 2, .., m} to

be j ammed is known and a set of possible locations Ni = {1, 2, ..., n} for the j amming devices

is given. Note than in the communication graph, V M. Let S~i := CE dijxy denote the
cumulative level of jamming at node i. Then node i is said to be j ammed if Si exceeds some

threshold value Ci. We say that communication is severed between nodes i and j if at least one

of the nodes is jammed. Further, let y : MZ1x Ms/ { 0, 1} be a surjection where yij := 1 if

there exists a path from node i to node j in the jammed network. Lastly, let z : Ms/ { 0, 1}) be a

surj ective function where ze returns 1 if node i is not j ammed.

The objective of the CONNECTIVITY INDEX PROBLEM (CIP) formulation of the WNJP is to

minimize total j amming cost subj ect to a constraint that the connectivity index of each node does

not exceed some pre-described level L. The corresponding optimization problem is given as:


(CIP) Minimize




xi E (0, 1}, jE N, I






where MllE R is some large constant.

Let v : M~/lx MZ/ {0 1} and v' : Mtlx Msl {0,( 1} be defined as follows:

0), otherwise,



1, if (i,: j) exists in the j ammed network,

0, otherwise.

With this, we can formulate an equivalent integer program as

(CIP-1) Minimize cx,


M~(1 ze) > Si Ci > -Mzei, Vie M Z,

Lemma 6. If CIP has an optimal solution then, CIP-1 has an optimal solution. Further, any

optimal solution x* of the optimization problem CIP-1 is an optimal solution of CIP.

Proof: It is easy to establish that if i and j are reachable from each other in the j ammed network

then in CIP-1, yij = 1. Indeed, if i and j are adj acent then there exists a sequence of pairwise

adj acent vertices:

{ (io, ii), ..., (im-1, im) },


where io = i, and im = j. Using induction it can be shown that yeimi = 1, V k = 1, 2, .., m.

From (4-16), we have that yiclt = 1. If yieai = 1, then by (4-17), yeoirct > Yiaik~ikikt = 1,

which proves the induction step.

The proven property implies that in CIP-1:

yei > connectivity index of i.


Therefore, if (x*, y*) and (x**, y**) are optimal solutions of CIP-1 and CIP correspondingly,


Vx* >' V x** (4-26)

where V is the obj ective in CIP- 1 and CIP.

As (x**, y**) is feasible in CIP, it can be easily checked that y** satisfies all feasibility

constraints in CIP-1 (it follows from the definition of myi in CIP). So, (x**, y**) is feasible in

CIP-1; thus proving the first statement of the lemma.

Hence from CIP-1,

V x** > V x* (4-27)

From (4-26) and (4-27):

V(x**) = V(x*). (4-28)

Let us define y7 such that

yij := 1 ++ j is reachable from i in the network jammed by x*.

Using (4-25), (x*, y) is feasible in CIP-1, and hence optimal. From the construction of y it

follows that (x*, y) is feasible in CIP. Relying on (4-28) we can claim that x* is an optimal

solution of CIP. The lemma is proved. O

We have therefore established a one-to-one correspondence between formulationS CIP

and CIP-1. Now, we can linearize the integer program CIP-1 by applying some standard

transformations. The resulting linear 0-1 program, CIP-2 is given as


(CIP-2) Minimize



ylo > v(4 V ,j .,M
y i k 1 ,j;Vi eM

xy E {0, 1}, Vje N,

, E {0, 1}, V i, je M,

In the following lemma, we provide a proof of equivalence between CIP-1 and CIP-2.

Lemma 7. If CIP-1 has an optimal solution then CIP-2 has an optimal solution. Furthermore,

any optimal solution x* of CIP-2 is an optimal solution of CIP-1.

Proof: For 0-1 variables the following equivalence holds:

The only differences between CIP-1 and CIP-2 are the constraints:

vUi = .zyze

v > < + z+ zj 2



Note that (4-38) implies (4-39) (1 .: > I z e + zy 2). Therefore, the feasibility region of

CIP-2 includes the feasibility region of CIP-1 This proves the first statement of the lemma.

From the last property we can also deduce that for all xl, x2 Such that xl is an optimal

solution of CIP-1, and x2 is optimal for CIP-2, that

V(xi) > V(Z2), (4-40)

where V(x) is the objective of CIP-1 and CIP-2.

Let (x*, y*, v/*, z*) be an optimal solution of CIP-2. Construct v//* using the following rules:

,, 1, if I + zf + z f 2 = 1,
vij (4-41)
0, otherwise.

v4 > 4;- (x*, y*, v /*, z*) is feasible in CIP-2 (yij > v4'f), hence optimal (the objective value is
V(x*), which is optimal). Using (4-41), (v /*, z*) satisfies:

Using this we have that (x*, y*, v//*, z*) is feasible for CIP-1. If x1 is an optimal solution of CIP-1

V(x ) < V(x*). (4-42)

On the other hand, using (4-40):

V(x*) < V(xi). (4-43)

(4-42) and (4-43) together imply V(xl) = V(x*). The last equality proves that x* is an optimal
solution of CIP-1. Thus, the lemma is proved. O

We have as a result of the above lemmata the following theorem which states that the

optimal solution to the linearized integer program CIP-2 is an optimal solution to the original
connectivity index problem CIP.

Theorem 12. If CIP has an optimal solution then CIP-2 has an optimal solution. Furthermore,

any optimal solution of CIP-2 is an optimal solution of CIP.

Proof: The theorem is an immediate corollary of Lemma 6 and Lemma 7. O

4.4 Deterministic Setup with Percentile Constraints

As we have seen, to suppress communication on a wireless network may not necessarily

imply that all nodes must be jammed. We might instead choose to constrain the connectivity

index of the nodes as in the CIP formulations. Alternatively, it may be sufficient to j am some

percentage of the total number of nodes in order to acquire an effective control over the network.

The latter can be accomplished by adding percentile risk constraints to the mathematical

formulation. Used extensively in financial engineering applications and optimization of stochastic

systems, risk measures have proven effective when applied to deterministic problems [120]. In

this section, we review two risk measures, namely Value at Risk (VaR) and Conditional Value at

Risk (CVaR) and provide formulation of the WNJP with the incorporation of these risk measures.

4.4.1 Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)

The Value-at-Risk (VaR) percentile measure is perhaps the most widely used in all

applications of risk management [103]. Stated simply, VaR is an upper percentile of a given

loss distribution. In other words, given a specified confidence level a~, the corresponding a~-VaR

is the lowest amount ( such that, with probability a~, the loss is less or equal to ( [121]. VaR

type risk measures are popular for several reasons including their simple definition and ease of


An alternative risk measure is Conditional Value-at-Risk (CVaR). Developed by Rockafellar

and Uryasev, CVaR is a percentile risk measure constructed for estimation and control of risks

in stochastic and uncertain environments. However, CVaR-based optimization techniques can

also be applied in a deterministic percentile framework. CVaR is defined as the conditional

expected loss under the condition that it exceeds VaR [168]. Figure 4-2 provides a graphical

representation of the VaR and CVaR concepts. As we will see, CVaR has many properties that
offer nice alternatives to VaR.

Let f (x, y) be a performance or loss function associated with the decision vector x C X C

IR", and a random vector in ye ERm. The y vector can be interpreted as the uncertainties that may

affect the loss. Then, for each x E X, the corresponding loss f (x, y) is a random variable having


Success a -CVaR

Figure 4-2: Graphical representation of Var and CVaR.

a distribution in IR which is induced by y. We assume that y is governed by a probability measure

P on a Borel set, say Y. Therefore, the probability of f (2, y) not exceeding some threshold value

( is given by

~(2, () := P {y| f (2, y) < (}. (4-44)

For a fixed decision vector 2-, ~((2, () is the cumulative distribution function of the loss associated

with 2-. This function is fundamental for defining VaR and CVaR [121].

With this, the a~-VaR and ca-CVaR values for the loss random variable f(2-, y) for any

specified as (0, 1) are denoted by g(xi) and #o(2-) respectively. From the aforementioned

definitions, they are given by

g,(2-) := min{( E R: ~((2, () > c0}, (4-45)


oa(2) := E{ f(2-, y)| f(2-, y) > g,(2-)}. (4-46)

Notice that the probability that f(2-, y) > g(0(2) is equal to 1 ca. Finally by definition, we have

that 40 (2-) is the conditional expectation that the loss corresponding to 2- is greater than or equal

to g(s(2) [162].

The key to including VaR and CVaR constraints into a model are the characterizations of

g,(:r) and #o(:r) in terms of a function F, : X x IR HR defined by

F,(:r, () := ( + E {max {f (:, y) (, 0} }. (4-47)
(1 a)

The following theorem, which provides the crucial properties of the function F, follow directly

from the paper by Rockafellar and Uryasey [162].

Theorem 13. As a function of (, Fo (:, () is convex and continuously di~ff~erentiable. The a~-C~aR

of the loss associated as ithr any xrE X can be determined f on the formula

#o(:r) = mmn Fa ir, (). (4-48)

hI this formula, the set consisting of the vahtes of (for 'I ithr the naininsun is attained, namely

A,(:r) = argmin Fa ir, (), (4-49)

is a nonenspty closed, bounded interval, and the ~-Va~R of the loss is given by

g,(:r) = left endpoint of A,(:r). (4-5 0)

In particular, it is aheays the case that

g,(:r) E argmin F,(:r, () and i' (:) = F,(:r, Co(:r)). (4-51)

This result provides an efficient linear optimization algorithm for CVaR. However, from a

numerical perspective, the convexity of Fair, () with respect to r and ( as provided by Theorem

13 is more valuable than the convexity of #o (:) with respect to r. As we will see in the following

theorem due to Rockafellar and Uryasey [163], this allows us to minimize CVaR without having

to proceed numerically through repeated calculations of #o (:) for various decisions r.

Theorem 14. Minimizing 40 (x) 0I ithr respect to x E X is equivalent to minimizing Fo (x, () over

all (x, () E Xx IR, in the sense that

mmn a (x) =mmn Fo (x, (), (4-52)
zEX (2,()EXxR

where moreover

(X*, (*) E argmin F,(x, () ++ x* E argmin #o(x), (* E argmin F,(x*, (). (4-5 3)
(z,()EX xR zEX (GIR

In the deterministic setting of the WNJP, we are not particularly interested in minimizing

VaR or CVaR as it pertains to the loss. Rather, we would like to impose percentile constraints on

the optimization model in order to handle a desired probability threshold. The following theorem

from [163] provides this capability.

Theorem 15. For any selection of probability thresholds asi and loss tolerances wi, i = 1, .. ., m,

the problem

mmn g (x) (4-54)
s. t.

,. (X) < Lo', for i = 1, m, (4-5 5)

where g is any objective function defined on X, is equivalent to the problem

mmn g(x) (4-56)
s. t.

Fo, (x, (4) < wi, for i = 1, m. (4-5 7)

Indeed,J (x*, (f ) solves the second problem ifandI only ifxr* solves the furst problem and

the inequality Fo, (x, (<) < wi holds for i = 1,. ., m.

Furthermore, ". (x*) < wi holds for all i = 1,. ., m. In particularfor each i such that

Fo, (x*, (*) = i, one has that ', (x*) = wi.

4.4.2 Percentile Constraints and the WNJP

In this section, we investigate the use of VaR and CVaR constraints when applied to the

formulations of the WNJP derived in Sections 4.3 and 4.4 above. As we have seen, risk measures

are generally designed for optimization under uncertainty. Since we are considering deterministic

formulations of the WNJP, we can interpret each communication node i EMi/ as a random

scenario, and apply the desired risk measures in this context.

We begin with the OPTIMAL NETWORK COVERING formulation of the WNJP. Suppose it

is determined that jamming some fraction a~E (0, 1) of the nodes is sufficient for effectively

dismantling the network. This can be accomplished by the inclusion of a~-VaR constraints in the

original model. Let y : Ms/ { 0, 1} be a surjection defined by

1, if node i is jammed,

Yi ~0, otherwise. (-8

Recall from Section 4.3 that Ni = {1, .., n} is the set of locations for the j amming devices,

and x is a binary vector of length a where xj = 1 if a j amming device is placed at location j.

Then to find the minimum number of jamming devices that will allow for covering a~ 101 1' of

the network nodes with prescribed levels of jamming Ci, we must solve the following integer


(ONC-VaR) Minimize cyxy (4-59)

ye > am,(4-60)
i= 1

dxj > Ciysi= ,,., (4-61)
Xj E {0,1} =1,,.,, (4-62)

yi E {0,1}, i 12,.m.(4-63)

Notice that this formulation differs from the ONC formulation with the addition of the a~-VaR

constraint (4-60). According to (4-61), if yi = 1 then node i is jammed. Lastly, we have from

(4-60) that at least 100 ( o' of the y variables are equal to 1.

The optimal solution to the ONC-VaR formulation will provide the minimum number of

j amming devices required to suppress communication on at least a~ 100% of the network nodes.

The resulting solution may provide coverage levels comparable to those provided by the ONC

model, while potentially reducing the number of jamming devices used. However, notice that the

remaining (1 a~) 100% of the nodes for which yi is potentially 0, there is no guarantee that they

will receive any amount of coverage. Furthermore, the addition of the m binary variables adds a

computational burden to a problem which is already NiP-hard.

We can also reformulate the CONNECTIVITY INDEX PROBLEM to include Value-at-Risk

constraints. Let p : MZ1 Z+ be a surjection where pi returns the connectivity index of node

i. That is, pi := Cm-,ypi, yid. Further let wU : Ms { 0, 1} be a decision variable having the

property that if It = 1, then pi < L. With this, the connectivity index formulation of WNJP with

VaR percentile constraints is given as

(CIP-VaR) Minimize cy xy (4-64)

p < et-+ (1 re M, i 1,,..,m,(4-65)

~ ~> am (4-66)
i= 1
Xj E {0,1}), j= 1,2,..., n, (4-67)

I, E {0,1}, i=i 12,..., m, (4-68)

pi E {0,1}, i= 1,2,..., m, (4-69)

where M~E R is some large constant.

Analogous to constraints (4-60)-(4-61), constraints (4-65)-(4-66) guarantee that at

least a~ 101 of the nodes will have connectivity index less than L. As with the ONC-VaR

formulation, there are two drawbacks of CIP-VaR. First, there is no control guarantee at all on any

of the remaining (1 a~) 101I' .; nodes for which I, = 0. Secondly, the addition of m binary

variables adds a tremendous computational burden to the problem. As an alternative to VaR, we

now examine formulations of the WNJP using Conditional Value-at-Risk constraints [162].

We first consider the OPTIMAL NETWORK COVERING problem. In order to put this into our

derived framework, we need to define the loss function associated with an instance of the ONC.

We introduce the function f : {0, 1}" x MZ/ HR defined by

f (x~,i):= Os xyda. (4-70)

That is, given a decision vector x representing the placement of the j amming devices, the loss

function is defined as the difference between the energy required to j am the network node i and

the cumulative amount of energy received at node i due to x. With this, we can formulate the

oNC with the addition of CVaR constraints as the following integer linear program:

(ONC-CVaR) Minimize cyxy (4-71)

(+~ ~ 1 a mi ydy-(,O< (4-72)
(1 a)mnCi
i= 1 j= 1
( E R, (4-73)

xj E {0, 1}, (4-74)

where Cmin is the minimal prescribed jamming level and dij is defined as above. The expression

on the left hand side of (4-72) is F,(x, (). Further, from Theorem 15 we see that constraint

(4-72) corresponds to having #o(x) < co = 0 [163]. Said differently, the CVaR constraint

(4-72) implies that in the (1 a~) 101I' .; of the worst (least) covered nodes, the average value of

f (x) < 0. For the case when Ci C for all i, it follows that the average level of jamming energy

received by the worst (1 a~) 101I' .; of nodes exceeds C.

The important point about this formulation is that we have not introduced additional integer

variables to the problem in order to add the percentile constraints. Recall, that in ONC-VaR

we introduced m discrete variables. Since we have to add only m real variables to replace

max-expressions under the summation and a real variable (, this formulation is much easier to

solve than ONC-VaR.

In a similar manner, we can formulate the CONNECTIVITY INDEX PROBLEM with the

addition of CVaR constraints. As before, we need to first define an appropriate loss function.

Recall that the definition of pi, the connectivity index of node i, is given as the number of nodes

reachable from i. Then can define the loss function f' for a network node i as the difference

between the connectivity index of i and the maximum allowable connectivity index L which

occurs as a result of the placement of the j amming devices according to x. That is, let f':

{0, 1}" x Ms1 Z be defined by

f '(x, i) := pi L. (4-75)

With this, the CIP-CVaR formulation is given as follows.

(CIP-CVaR) MinimnizeC cyxy (476)

S+ 1max ,i ; L -(,O} (4-77)
(1 -a~mi= 1
pi E Z, (4-78)

( E R, (4-79)

where pi is defined as above. As with the previous formulation, the expression on the left-hand

side of (4-77) is F,(x, () from (4-47). Furthermore, we have from from Theorem 15 that

(4-77) corresponds to having #o (x) < co = 0. This constraint on CVaR provides that for the

(1 a~) 10a' .; of the worst cases, the average connectivity index will not exceed L. Again, we

see that in order to include the CVaR constraint, we only need to add (m + 1) real variables to the

Table 4-1: Optimal solutions using the coverage formulation with regular and VaR constraints.
Optimal Solutions Regular Constraints VaR Constraints
Number of Jammers 6 4
Level of Jamming 101II' V nodes 101 1' for II. of nodes,
85% (of reqd.) for 4% of
CPLEX Time 0.81 sec 0.98 sec

problem. Computationally, CVaR provides a more conservative solution and will be much easier

to solve than the CIP-VaR formulation as we will see in the next section.

4.5 Case Studies and Algorithms

In order to demonstrate the advantages and disadvantages of the proposed formulations for

the WNJP, we will present two case studies. The experiments were performed on a PC equipped

with a 1.4MHz Intel PentiumB 4 processor with 1GB of RAM, working under the Microsoft

Windows@ XP SP1 operating system. In the first study, an example network is given and the

problem is modeled using the proposed coverage formulation. The problem is then solved exactly

using the commercial integer programming software package, CPLEX Next, we modify the

problem to include VaR and CVaR constraints and again use CPLEXB to solve the resulting

problems. Numerical results are presented and the three formulations are compared. In the

second case study, we model and solve the problem using the connectivity index formulation. We

then include percentile constraints re-optimize. Finally, we analyze the results.

4.5.1 Coverage Formulation

Here we present two networks and solve the WNJP using the network covering (oNC)

formulation. The first network has 100 communication nodes and the number of available

j amming devices is 36. The cost of placing a j amming device at location j, cj is equal to 1 for all

locations. This problem was solved using the regular constraints and the VaR type constraints.

Recall that there is a set of possible locations at which j amming devices can be placed. In these

examples, this set of points constitutes a uniform grid over the battlespace. The placement of the

jamming devices from each solution can be seen in Figure 4-3. The numerical results detailing

the level of jamming for the network nodes is given in Table 4-1. Notice that the VaR solution

*I *

* I ,I I

*I *

Figure 4-3: Case study 1. The placement of jammers is shown when the problem is solved using
the original and VaR constraints.

Table 4-2: Optimal solutions using the coverage formulation with regular and VaR, and CVaR
Opt Solns Reg (all) VaR (.9 conf) CVaR (.7
# Jammers 9 8 7
Jamming Level 101II' V nodes 10I o 1'. 100% for 57%
of nodes, of nodes,
72% for 10% 90% for 20%
of nodes ofnodes,
76% for 23%
of nodes
CPLEXB Time 15 sec 15h 55min 41 sec

called for 33% less jamming devices than the original problem while providing almost the same

jamming quality.

In the second example, the network has 100 communication nodes and 72 available

jammers. This problem was solved using the regular constraints as well as both types of

*Network nodes

m Regular Constraints
Ar VAR Constra nt




*1 *

.C *
A1 \1

*Network Nodes
X Re gula r C onstrali nts
*) CVAR Constraint
At VA R C onstraint

Figure 4-4: Case study 1 continued. The placement of jammers is shown when the problem is
solved using VaR and CVaR constraints.

percentile constraints. The resulting graph is shown in Figure 4-4. The corresponding numerical

results are given in Table 4-2.

In this example, the VaR formulation requires 1 1% less jamming devices with almost the

same quality as the formulation with the standard constraints. However, this formulation requires

nearly 16 hours of computation time. The CVaR formulation gives a solution with a very good

j amming quality and requires 22% less j amming devices than the standard formulation and 1 1%

less devices than the VaR formulation. Furthermore, the CVaR formulation requires an order of

magnitude less computing time than the formulation with VaR constraints.

4.5.2 Connectivity Formulation

We now present a case study where the WNJP was solved using the connectivity index

formulation (CIP). The communication graph consists of 30 nodes and 60 edges. The maximal

number of jamming devices available is 36. We set the maximal allowed connectivity index of

any node to be 3. In Figure 4-5 we can see the original graph with the communication links prior

to j amming. The result of the VaR and CVaR solutions is seen in Figure 4-6. The confidence

A *


rA *

Figure 4-5: Case Study 2:

Original graph.

Figure 4-6: A comparison of the percentile constrained solutions. In both cases, the triangles
represent the placement of jamming devices. (a) VaR Solution. (b) CVaR Solution.

level for both the VaR and CVaR formulations was 0.9. Both formulations provide optimal

solutions for the given instance. The resulting computation time for the VaR formulation was 15

minutes 34 seconds, while the CVaR formulation required only 7 minutes 33 seconds.

4.6 Concluding IRemarks

In this chapter we introduced the deterministic WIRELESS NETWORK JAMMING PROBLEM

and provided several formulations using node covering constraints as well as constraints on the

connectivity indices of the network nodes. WNe also incorporated percentile constraints into the

derived formulations. Further, we provided two case studies comparing the twvo formulations with

and without the risk constraints.

With the introduction of this problem, we also recognize that several extensions can be

made. For example, all of the formulations presented in this chapter assume that the network

topology of the enemy network is known. It is reasonable to assume that this is not always the

case. In fact, there may be little or no a priori information about the netwvorki to be jammed.

In this case, stochastic formulations should be considered and analyzed. Thi s brings us to the

topic of the next I-hi.rl-pi in which we consider the case when no information is known about the

network to be jammed other than its relative location inside a planar region.


5.1 Introduction

This chapter describes a problem of interdicting/j amming communication networks

in uncertain environments [44]. Interdiction of communication networks is an important

application, but as previously mentioned, has not been intensively researched despite the vast

amount of work on optimizing telecommunication systems [155]. Most papers on network

interdiction are about preventing j amming and analyzing network vulnerability [68, 134]. To

our knowledge, the only literature on network interdiction involving optimal placement of

jamming devices is the work of Commander et al. [45] (presented in Chapter 4) in which several

mathematical programming formulations were given for the deterministic WIRELESS NETWORK

JAMMING PROBLEM. The only other thoroughly studied cases are problems of minimizing the

maximal network flow and maximizing the shortest path between given nodes via arc interdiction

using limited resources. Cormican et al. [49], Israeli et al. [110], and Wood [174] studied

stochastic and deterministic cases and suggested efficient heuristics. A similar setup but with a

different obj ective was recently studied by Held in 2005 [95].

This problem is particularly important in the global war on terrorism as improvised

explosive devices (IEDs) continue to plague the coalition forces in Iraq. These homemade

bombs are almost always detonated by some form of radio frequency device such as cellular

telephones, pagers, and garage door openers. The ability to suppress radio waves in a region

will help prevent casualties resulting from IEDs. Furthermore, since most situations arise in

military battlefield scenarios, exact information about the topology of the adversary's network

is unknown. Thus, deterministic network interdiction approaches have limited applicability. In

this case, a stochastic approach involving some risk measure for evaluating the efficiency of the

jamming device placement may be helpful. However, choosing an appropriate risk measure is a

challenging problem in its own right. In this chapter, we consider an extreme case where there is

no a priori information about the topology of the network to be j ammed. The only information

used in our approach is a bounding area, containing the communication network.

The organization of this chapter is as follows. Section 5.3 gives a formal description of

the problem and the j amming model. We derive bounds and prove a convergence result for the

case of complete uncertainty in Section 5.4. Here we also demonstrate the advantage of the

proposed method compared to the simplified case which does not account for the cumulative

effect of the j amming devices. In Section 5.5 we present a randomized local search and illustrate

its effectiveness by using the bounds derived in the previous section. Section 5.6 provides some

concluding remarks.

5.2 Descriptions, Assumptions, and Definitions

In general, the problem of jamming a communication network is to determine the minimum

number of jamming devices required to interdict or suppress functionality of the network.

Starting with this general statement, more specific ones can be obtained by considering various

types of jamming devices and interdiction criteria. Depending on the given information about

the communication nodes and the network topology, stochastic or deterministic setups can

be constructed [45]. Below we provide assumptions and basic definitions of the considered


We consider radio-transmitting communication networks and j amming devices operating

with electromagnetic waves. We assume that the jamming devices have omnidirectional antennas

and emit electromagnetic waves in all directions with the same intensity. We also assume that

jamming power decreases reciprocally to the squared distance from a device.

Definition 12. A point (conanunication node) X is said to be jananed or covered if the

cumulative energy received fion; all ja~nmning devices exceeds some threshold vahte E:

C~> E, (5-1)
71 R(X, i)

where A E R and R(X, i) represents the distance fr~om X to ja~nmning device i. This condition

can be rewritten as:

1 _1 (5-2)
SR2(X, i) L2

where L= .

The latter inequality implies that a j amming device covers any point inside a circle of radius

Definition 13. A connection (arc) beaveen avo conanunication nodes is considered blocked ifany

of the avo nodes is covered.

Usually, interdiction efficiency is determined by a fraction of covered nodes and/or arcs.

More complicated criteria used are based on the amount of information transmitted through the

network or the length of the shortest path between pairs of nodes. We do not consider a specific

criterium because we are interested in the case of complete uncertainty. Thus, we are assuming

that we have no knowledge of the network topology, including information about the node


5.3 Problem Formulation

If we ignore the cumulative effect of the j amming devices, then the problem reduces to

determining the optimal covering of an area on a plane by circles. This covering problem was

solved in 1936 by Kershner [113]. The current chapter shows that accounting for the cumulative

effect of all the devices can lead to significant losses in costs, i.e. required number of jamming


Since we assume no information is known about the network to be jammed, the only

reasonable approach is to cover all points in some area known to contain the network. This

approach would also be appropriate when some information about the network is available, but is

potentially inaccurate.

We consider a case when a communication network is located inside a square. However,

all of the following theorems can be formulated for a more general case. For example, to

i i i i I

Figure 5-1: Uniform grid with j amming devices

obtain results when the network is contained inside a rectangular region in the plane, the only
modification required to the calculations is an appropriate updating of the summation bounds.
An optimal covering is one which contains the minimum number of jamming devices that
jam all points in the particular area of interest. However, finding a globally optimal solution for
the general problem is difficult [45]. Therefore, we consider a subproblem of covering a square
with j amming devices located at the nodes of a uniform grid. The solution to this problem will
provide a feasible solution (optimal in certain cases) to the general problem. Suppose the grid
step size is R. If the length of a square side a is not a multiple of R, then we cover a bigger
square with a side of length R([ ] + 1). See Figure 5-1 for an example. The optimal solution
in the considered problem is a uniform grid with the largest possible step size which covers the
square. The problem remains non-trivial, even for this simplified setup.
Lemma 8. For any covering of a square 0I ithr a uniform grid, a point which receives the least
amount of jamming energy lies inside a corner grid cell (see Figure 5-2).

Proof: Consider a corner cell So and an arbitrary non-corner cell Si. We prove that for any point
P E Si, there is a corresponding point P' E So such that E(P) > E(P'), where E(X) is the
cumulative j amming energy from all devices received at point X.
Let P' be a symmetric correspondence of point P inside So. Here, symmetry implies that
P and P' are equidistant from the sides of their respective cells. We split the square into the


least covered point


Y P'

Figure 5-2: The least covered point is shown in the lower left grid cell.

four rectangles A, B, C, and D, where A is the rectangle containing cells So and Si (see Figure

Figure 5-3: Square Decomposition

5-3). Denote the other two corner cells of rectangle A by C1 and C2. Let also Ti and T2 be points

inside C1 and C2 TOSpectively, such that TIPT2F' is a rectangle with sides parallel to the sides of

the square as in Figure 5-4. Using symmetry we get the following relations:

E(P', A)

E(P', B) < E(TI, B)

E(P', D) < E(T2, D)

E(P', C)

= (P, A),

= (P, B),

= (P, D),

< E(P, C),





Figure 5-4: Equivalent Points

where E(X, I) is the cumulative j amming energy from all devices inside rectangle I received by

point X. Relations (5-3) (5-6) imply

E (P') = E (P', A) + E (P', B) + E (P', C) + E (P', D)

< E(P, A) + E(P, B) + E(P, C) + E(P, D) (5-7)

=E(P) ,

and the lemma is proved. O

5.4 Upper and Lower Bounds

Below we formulate theorems for upper R and lower R bounds for the optimal grid step size

R* : R < R* < R. In all formulated theorems, we consider covering a square with side length a.

Theorem 16. Thze unique sohttion of the equation

1a 1
2R2~ R Llt; I 2 58

is a lower bound R for the optimal grid step size R*.

Proof: In Lemma 8, we proved that the least covered point lies inside a corner cell. Consider

now a grid with step size R. Without the loss of generality, let P(.ro, Yo) be a point inside the

,.u Pi

Figure 5-5: Cumulative emanation of jamming devices.

bottom left corner cell as shown in Figure 5-5. Il, I2, and I3 are cumulative j amming energy

received at P by jamming devices located in regions C, A, and B correspondingly. Similarly, I4

is the j amming energy from the j amming device located at the bottom left node O. With this, the

jamming energy received at point P is calculated through the expression

E(P) = II + I2 3 4 I, where
T-1 T-1

I~CC (R -xo + i R)2 90 o R2
i=0 j=0

i= o (R xo i R) 2 Y2,

IjCX~ (R yo j -R)2


Notice that we can estimate I2 3 4 S

T-l1 2 l 1
G~hPC R2( i2 + 2 R2 0 1 1+X2






Figure 5-6: Integral Lower Bound.

This follows from the fact that

f i)> f (x)dx, (5-16)
where f (x) is a decreasing function. This property can be easily established geometrically.

Notice in Figure 5-6 that the left side of inequality (5-16) represents the shaded region in the

figure, while the right side represents the area under f (x). Continuing from (5-15) above we

/ T 1 (1 + x)2

=arctan(T + 1) -

2 T14
S4 T+1


Here and further, we use the inequalities given below:

arctan(x) < x, O
arctan(x) > x -- -,O < x< 1.



Now combining (5-15) and (5-17), we obtain

I23> I-2 -
SR2 4

- .
/ J


We also have the following approximation for I4 which follows clearly



For estimating II we use a property similar to (5-16), but in a higher dimension. Namely,


i=0 j=0

.N+1 *N+1
0 J 0

f (x, y).1<. Li


where as above, f (x, y) is a decreasing function of x and y. Using this inequality, we derive the

following approximation for Il.

/ dxdy
(R-x+ x -R)2 _9 (R. @2)
/ + + dxdy

R2 1f J1 2 2'



.1 T1

T+1 dxdy /


> '

T 1 dl

T 1
arctan dx

T 1
arctan dx

- -



1 +



xn( 1
SIn(T + 1)

In(T + 1)

x~ T+1i

1 +

2 1


arctan (-


T+1 > i

f1+1 oX

Combining this result with (5-23) we have

1 1
li > I( +1 -21 (5-25)

Summing (5-15), (5-21), and (5-25) we obtain an overestimate of the total coverage at

point P. That is

1 x2 xr 2 1\
E(P) > ~- -- n(T 1)P -t 2
R2 2T+12 T+1 2/

In( + )i +\2 (5-26)
2R2f 2 2 2u+)+-)

> -1 xI + 1 ) iT- .i)>-(-7
2R2 C2

Since f (R) is monotonically decreasing on (0, +oo), the largest R satisfying the above inequality

is the unique solution R of the equation

f (R) = 2.(5-28)

Thus, a uniform grid with step size R jams any point P inside a corner cell. According to Lemma

8, the grid jams the least covered point in the square implying that the whole square is jammed.
Thus we have the desired result. O

Since the function f (R) =k (x In(~ 1) + i 3) is monotonic, equation (5-8) can be

easily solved using a numerical procedure such as a binary search. Therefore, using (5-8), we can

obtain a step size R such that the corresponding uniform grid covers the entire square. Further, it

is easy to see that the number of jamming devices in the grid does not exceed

NI +2 (5-29)

Table 5-1: Comparing ~-for various values of k.

102 2.44 2.3
10 3.54 4.8
106 4.40 7.5
108 5.14 10.2

A more straightforward solution of the initial problem could be based on the property that a

jamming device covers all the points inside a circle of radius L as mentioned in Definition 12.

Using that, we could reduce the problem to finding the optimal covering of a square with circles

of radius L. A direct result from [1 13] (that was mentioned in [134]) is that in the limit, the

minimum number of circles to cover an area a2 is

N2 = (5-30)

To compare the approaches, we consider the ratio

N2 R2 1

(5-3 1)

where L x ndk Using these substitutions, equation (5-8) can be rewritten in terms of

variables x and k as follows

I 1 x- 3 = 2. (5-32)

By solving (5-32) for different values of k, one can find corresponding values of x and .

To evaluate the advantage of the uniform grid approach over the naive one, we provide some

computational results in the Table 5-1. From the table, we see that as k increases, the advantage

of using our approach becomes more significant. In fact, it can be proved that lime-m~ = o

This will follow as a corollary of Theorem 18.

To establish the quality of the lower bound rigorously, we need to first establish a similar

result for an upper bound. This follows in the next theorem.

Theorem 17. Thze unique solution of the equation

1 x 2a1 xr 19 1
In + -+ + (5-33)

is an upper bound R of the optimal grid step size R*.

Proof: Let P(xo, yo) be the least j ammed point, that lies inside a corner cell according to Lemma

8. Without the loss of generality, as in the proof of Theorem 16, we assume that P is inside

the bottom left corner cell. The j amming energy received at point P is calculated through the

expressions (5-9) (5-14). Since P is the least covered point, the following inequality holds.

E(P)< E x= y= 0 = I' + I' + I' I', where (5-34)
T-1 T-1
I' =(5-35)

I' = ( )2,(5-36)


I' = 1n~j ) ? (5 -37)

I' = .(5-3 8)

I' nd cn be, estimatedl thr~roug integrals similarly to the techniques used in the proof of

Theorem 16. The following inequality holds

f= (i) f(x;d, (5-39)

where f (x) is a decreasing function. This property can also be proven geometrically. Figure 5-7

represents a graphical interpretation of this relation. The left side of the inequality is represented

by the shaded area. The right side of (5-39) is the area under f (x). With this property we have

Figure 5-7: Integral Upper Bound.

from (5-36) that

1 r-
I'aa < +




Furthermore, using inequalities (5-18) and (5-19), we see that (5-37) is estimated by

I' <

(~2 R+zR2
3R22 R22 arctan j
3R2 2 2

2 21 a

2T1 24T31

12T3 '


R2 3


To estimate I' a pr~nropert similarr~l to ~ (539cn be used.l This inmneulityr~ is givn by

i= 1 j= 1 0J O



where f (x, y) is a decreasing function of x and y. With the above inequality,

O -

1 r-
I' < +I
1 2) 2 0


(~2 R+zR2




Jo (4 + x ) t R)2 2 )
C 1 d(X 2 1
"2 R2 002

:Y 12


arctan(2) + arctan T :
2T1 rcta 2/(

1 1 2

!4T3 2 2T 1

Irctan 2T-1

2~Th 1)

C = 2 arctan(2T)

S- 2 arctan





The double integral in (5-43) is bounded as follows

d( + x)2 2

J1 {2 2

a r c t a n


43 T -1

- arctan Tj\ dt -
1\~ \

S6(T' -1 )21

211 1 l
t\ t 3t3



5 1

6(T 1)2

In(2T 1) -

+ +
3 6iT 12T2

36T3 IT -

12(T ~)2'

xT 20 5 1
< n(2T 1) +; +

Combining the results from (5-43), (5-44), and (5-45) gives the overestimate for 7( a

7(< I2 ~n(2T

S12(T -1 )21

xr 16 5
1) +I +
2 3 6iT T


Jo Jo

- rcand

Recall equation (5-J 34)sated E(P) < 7( + 7 + If + Id. So using the expression for If given
in (5-38) and the overestimates for 7(, I(, and I' de-rived in equatio~ns1 (5-46) (5-0) and (5-41) l

respectively, we obtain

1 x1 xr 19\
E(P < -In2T- 1 + (5-47)
EP R2 n~ qn21 6) T 2 3/
Finally, if we let T = [ ] 1 < + ,w e

1 x 2a1 xr 19\
E(P)< In1 -+ + (5-48)
R2 2a R> 6 ( 1) 2 3
The function f(R) = In,! 1)- is monotone, hence the equation

f (R) = ~zhas a unique solution R. Equation (5-48) implies that a grid with step size R does

not cover the entire square. That is, there exists at least one point P that remains uncovered.

Thus R is an upper bound for the optimal grid covering problem. Since the optimal grid step size

R* < R, the theorem is proved. O

In Figure 5-8, we see an example in which we are covering at 40 x 40 square and the

required j amming level at each point is 3.0 units. In part (a), we see the coverage associated with

the required number of devices from the lower bound of Theorem 16. In this case, 202 = 400

jamming devices are used to cover the area. Notice that there are no holes in the region. This,

together with the scallop shell outside the bounding box indicates that all points within the region

are covered. In part (b), we see the coverage corresponding to the placement of the j amming

devices on a uniform grid according to the upper bound of Theorem 17. Here, the required

number of devices is 192 = 361. Notice the holes located at the four corners of the region

indicating that these points are uncovered. This validates the theoretical results obtained in

Theorem 16 and Theorem 17.

Now that we have established both upper and lower bounds for an optimal grid step size, we

can determine the quality of the bounds. The result is obtained in the following theorem.

Theorem 18.

lim = 1, (5-49)

155 0 1 0 0 1 0 2

10> -b

Fiue5-8 oprs o h oe n pe ons a h oea o h mig
0-ie r lcdacrigtotelwrbudfo hoe 2 h oa ubrojamn

Figure 5-8: Comparisonse of2 th lowe ade s upper bounds. (a)c The coverage poft whe n jamming

where R and R are bounds obtained fr~om equations (5-8) and (5-33), correspondingly.

Moreover, the following inequality holds:

R c
1 <- < 1 + (5-50)
R- In~a '

for constants Me E R, ce ER, such that R > M~.

Proof: By letting x := and y-- := ,,, equaion (5-8 and 2 (5-33 can be nrespetivly~ mreritten as

a = L~ x ej ("+ -' 1, and (5-51)

S 2a 19 x L-
In = 2 (5-52)
2 Ly3 2 6(a + L y)'

To prove the theorem, we need to show that

limn = 1 (5-_53)

where x > 0 and y > 0 are solutions of (5-51) and (5-52), correspondingly. From (5-52), we


SI2 L-ya

+ 1>u y2_1 where
19 xr
C = and




ay > e: 2 v-c )

From (5-51) and (5-56) we see that

x e"(22+c2) C3

1 6- (y -01)

-) 1 where
C2 and
C3 = 6-1.




Since y L and x L are upper and lower bounds, correspondingly, the following relation holds

> 1.


With (5-51) and (5-60) above, we can also conclude that

lim x

00 and lirn y


For all Me E R, where M~ > zC, there exists Q E R such that (5-57) can be reduced to

< Q -e 2("2-Y2), and y > M.


In(Q) the following, inequality, holds

Moreover, for c


1 < ,,and y > M.


Assume for the sake of contradiction that the inequality in (5-63) does not hold for some

(r*, y*). That is assume that ( )2

- 1> -. Using (5-62) we have



which contradicts (5-60).

Applying (5-60) and (5-63) we get

1 < < and y > M~. (5-65)

Letting a tend to 00 and taking (5-61) into account, we see that in fact

lim = 1.(5-66)

Finally, by using (5-65) and (5-51), the following relation can be obtained

y k
1 < < 1 +(5-67)
x In(a>)

for some constant k E IR, when y > M~. Thus, the theorem is proved. O

5.5 Heuristic for Uncertain Jamming

Here, we describe the implementation of a randomized local search heuristic for the case

of jamming under complete uncertainty. Recall that the subproblem for which the bounds

in Theorem 16 and Theorem 17 we derived place n jamming devices, where n is a perfect

square. The obvious drawback of this technique is the situation where for example R requires 25

jamming devices and R calls for 16, and the optimal solution to the general problem is 18. Using

the uniform grid approach will require nearly 1 1II' more devices than are needed to cover the


Pseudo-code for the local search is given in Figure 5-9. The heuristic takes as input the

size of the region containing the network (region). The number of jamming devices required to

cover the area of region by the lower bound on the grid step (upper bound on j amming devices)

derived in Theorem 16 (ubJammers) is the second input parameter. In line 1, the optimal solution

(X*) is set to ubJammers.

The while loop from lines 2-12 is where the local optimization takes place. In line 3, the

jamming devices are randomly scattered within the square region known to contain the network.

Next in the while loop from lines 5-8, those points which are receiving the least amount of

procedure randLocalSear ch(region, ub Jammers)
S X* <- ubJammers
2 while stoppingCriteria = FALSE do
3 randScatter(region,X*)
4 local~pt = FALSE
5 while local~pt = FALSE do
6 P <-leastJammedPoints (region)
7 move~ammers(p)
8 end
9if all Jammed = TRUE then
to X* <- X*- 1
11 end
12 end
13return X*
end procedure randLocalSearch

Figure 5-9: Pseudo-code for the randomized local search for uncertain j amming.

j amming energy are assigned to the set p. Then, the j amming devices are moved along a gradient

towards the points in P until these are points are covered. Several methods are available for

the function moveJammers including the method of steepest descent [17] or the more efficient

method of conjugated gradients [91, 92]. The heuristic then determines if all points have been

j ammed. If this is the case, then in line 9 we decrement the number of jamming devices by one

and return to line 2. If all points are not j ammed, we repeat the loop until either all points are

covered or until a stopping criteria is met in which case we exit the while loop. The final value of

X* is returned as the solution in line 13.

An example can be seen in Figure 5-10. For this example, a point requires 3 units of

jamming energy before it is declared to be jammed. Figure 5-10(a) represents the placement

of the j amming devices according to the optimal uniform grid solution from Theorem 25. In

this case, 400 devices are required. In Figure 5-10(b) we see the associated coverage from this

solution. The scallop shell around the bounding box containing the network indicates that in fact,

the entire area is jammed, but perhaps more devices are used than are necessary. In subfigure

(c), we see the placement of the 298 j amming devices according to the heuristic solution. Notice

gu2 -5 10 -5 0 5 10 15 20


15 *



-15 *

-20 -15 -10 -5 0 5 10 15 20 25


-20 -15 -10 -5 0 5 10 15 20 25

Figure 5-10: Example of heuristic versus uniform placement. (a) Device placement on uniform
grid. The total number of jamming devices required is 400. (b) Coverage display of uniform
placement. (c) Heuristic jammer placement. The total number of required devices is 298. (d)
Heuristic coverage plot.

in Figure 5-10(d) that the coverage outside the bounding box is reduced significantly while still

jamming all points in the region. The heuristic reduces the required number of devices by 25.5'

Numerical results for several regions with various required jamming levels can be seen in

Table 5-2. In this table, we list, the side length of the region (| a|), the required j amming level

(L), the number of jamming devices required by the upper and lower bounds computed using

Theorem 16 and Theorem 17. Next, we list the required j amming devices corresponding to the

optimal grid step size which was determined using a binary search with the upper and lower



6.1 Introduction

In many situations, multiple "agents" work together to achieve a shared goal. Cooperation

between the agents is important to improve the efficiency and effectiveness by which their

goal is reached. In information systems this idea also holds. That is, in wireless networks,

groups of agents are often employed to perform a number of cooperative tasks including the

synchronization of information among a set of users, and the accomplishment of missions

in remote areas. In such situations, it is useful to maintain collaboration among the agents

performing the cooperative tasks in order to maximize the probability of success.

Communication is an important measure of collaboration between entities involved in a

mission. It allows different agents to perform the set of tasks that have been planned, and at the

same time to implement changes in the case that an unexpected event occurs. Moreover, high

communication levels are necessary in order to perform complicated tasks, where several agents

must be coordinated. We describe in this section the main concepts found in the literature related

to optimizing communication time in ad hoc network systems.

One of the main difficulties concerning the maintenance of communication is an ad hoc

network is determining the location of agents at a given moment in time. Several methods have

been proposed for improving localization in this situation. Moore et al. [132], for example,

presented a linear time algorithm for determining the location of nodes in an ad hoc network in

the presence of noise. Other algorithms for the same problem have been suggested by Capkun et

al. [29], Doherty et al. [60], and Priyantha et al. [153].

While such algorithms can be useful in determining the correct location of nodes, they are

only able to provide information about current positions, and are not meant to optimize locations

for a specific obj ective. Packet routing, on the other hand, has been previously studied with the

goal of optimizing some common parameters, such as latency, cost of the resulting route, and

energy consumed. For example, Butenko et al. [25] proposed a new algorithm for computing a

backbone for wireless networks with minimum size, based on a number of related algorithms for

this problem [23, 33, 129].

Another problem involving the minimization of an obj ective function over all feasible

positions of agents in an ad hoc network is the so-called location error minimization problem. In

the LOCATION ERROR MINIMIZATION PROBLEM, given a set of measurements of node positions

(taken from different sources), the goal is to determine a set of locations for wireless nodes such

that errors in the given measurements are minimized. This problem has been formulated and

solved using mathematical programming techniques, by the use of a relaxation for the general

problem into a semi-definite programming model [19, 20, 46, 165].

There are many applications of this described system. These include situations where

communication in a region is required, but no topologically fixed transmission system exists.

Specific examples include emergency/rescue operations, disaster relief, battlefield operations,

and BluetoothB systems [137]. In each of these examples the goals and obj ectives are fixed

in advance and communication is important for the attainment of these goals. The current

technologies used in these type of applications allow improved communication systems that rely

on ad hoc wireless protocols. However, it is a combinatorial problem to decide how to maintain

communication for the maximum possible time, when faced with the inherent restrictions of

wireless systems.

Advances in wireless communication and networking have lead to the development of new

network organizations based on autonomous systems. Among the most important example of

such networks systems are mobile ad hoc networks (MANETs). MANETs are composed of a

set of loosely coupled mobile agents which communicate using a wireless medium via a shared

radio channel. Agents in the network act as both clients and as servers and use various multi-hop

protocols to route messages to other users in the system [137, 141]. Unlike traditional cellular

systems, mobile ad hoc networks have no fixed topology. Moreover, in a MANET the topology

changes each time an agent changes its location. Thus, the communication between the agents

depends on their physical location and their particular radio devices.

Interest in MANETs has surged in the recent years, due to their numerous civilian and

military applications [155]. MANETs can be successfully implemented in situations where

communication is necessary, but no fixed telephony system exists. Real applications abound,

especially when considering adversarial environments, such as the coordination of unmanned

aerial vehicles (UAVs) and combat search and rescue groups. Other examples include the

coordination of agents in a hostile environment, sensing, and monitoring. More generally,

the study of protocols and algorithms for MANETs is of high importance for the successful

deployment of sensor networks which are themselves composed of a large number of

autonomous processors that can coordinate to achieve some higher level task such as sensing

and monitoring.

The lack of a central authority in MANETs leads to several problems in the areas of routing

and quality assurance [25]. Many of these problems can be viewed as combinatorial in nature,

since they involve finding sets of discrete obj ects satisfying some definite property, such as for

example, connectedness or minimum cost. Among the challenging problems encountered in

MANETs, we can cite routing, or path planning as one of the most difficult to solve, because of

the temporary nature of communication links in such a system. In fact, as nodes move around,

they dynamically define topologies for the entire network. In such an environment, it is difficult

to determine if two nodes are connected, since any of the intermediate nodes may leave the

network at any time.

This scenario makes clear the importance of close coordination among groups of nodes

if a definite goal needs to be attained. If at all possible, a plan must be devised such that

communication among nodes is maintained for as long as possible. With this obj ective in

mind, we study in this chapter, a problem involving the coordination of wireless users involved

in a mission of tasks that requires each user to go from an initial location to a target location.

The problem consists of maximizing the amount of connectivity among the set of users,

subj ect to constraints on the maximum distance traveled by users, as well as restrictions on

what types of movement can be performed. The resulting problem, called the COOPERATIVE


defined in a later subsection.

This chapter is organized as follows. Section 6.2 begins with a brief review of some of

the previous work in the areas related to cooperative communication in wireless systems. Then

we derive the discrete version of the CCPMANET, referred to as CCPMANET-D. Specifically,

we formulate the problem as a combinatorial problem on a graph and provide an integer

programming formulation. In Section 6.3, we provide a suite of heuristic algorithms for

the problem beginning with a simple construction algorithm and culminating with the

implementation of a GRASP with path-relinking for the CCPMANET-D. Computational results are

presented and the methods compared in Subsection 6.3.6. In Section 6.4 we derive a continuous

formulation called CCPMANET-C. A continuous version is more likely to model real-world

scenarios as we no longer rely on the underlying graph structure and the agents are free to move

subj ect to kinematic constraints. Finally, concluding remarks and future research ideas are

presented in Section 6.5.

6.2 Discrete Formulations (CCPMANET-D)

As mentioned above, ad hoc networks represent an extremely active area of research [155].

Several problems related to routing, power control, and accurate position update, have been

studied in the last few years [136]. In terms of routing, one of the main problems in ad-hoc

networks is the computation of a network backbone. The obj ective is to find a subset of nodes

with a small number of elements that can be used to send routing information. The use of such

a structure is useful to simplify the management tasks required by a routing protocol. The

backbone computation problem can be modeled as a CONNECTED DOMINATING SET (CDS)

problem. Here, the obj ective is to find a set of minimum size forming a connected backbone, with

the additional property that each network client can directly reach this set. The CDS problem,

which can be modeled using unit graphs, has several approximation algorithms [23, 26, 33], all

of which are based on approximation properties of the MAXIMUM INDEPENDENT SET problem

on planar, unit graphs [12]. The use of discrete optimization techniques to maximize connectivity

in ad hoc systems was is a relatively new idea which was put forth by Oliveira and Pardalos in

[137]. We now present some discrete formulations for the CCMPANET-D.

Consider a graph G = (V, E), where V = {vl, v2, v,) TepreSents the set of candidate

positions for the wireless agents. Suppose that a node in G is connected only to those nodes that

can be reached in one unit of time. Let U represent the set of agents, S = {sl, s2, **>s|U|} C V

represent the set of initial positions, and D = {dl, d2, *, d|U|} C V the set of destination nodes.

Let NV(v) C 2V, for v E V, represent the set of neighbors of node v in G. Given a time horizon

T, the obj ective of the problem is to determine a set of routes for the agents in U, such that each

agent ui E U starts at a source node si and finishes at the destination node di E D after at most T

units of time.

For each agent u E U, the function pt : U V returns the position of the agent at time

te { 1, 2, .. ., T}, where T is the time limit by which the agents must reach their destinations.

Then at each time instant t, an agent E U can either remain in its current location, i.e. pt-1(u),

or move to a node in NV(pt-1(u)).

We can represent a route for an agent E U as a path P = {vl, v2,. ., Uk} in G where

vi1 = s,, I I = d,, and, for is { 2,..., k}, E N(I _1) U {< }. Finally, if {p,)} ,l is the set

of traj ectories for the agents, we are given a corresponding vector L such that 4 is a threshold

on the size of path pi. This value is typically determined by fuel or battery life constraints on the

wireless agents.

We now have to decide what the actual measure of connectivity amongst the agents on their

traj ectories will be. Obviously, the best possible situation would be the case when all agents in

the network are linked. However, in ad hoc systems, this is unlikely due to limits on power and

fuel. As noted in [137], one possible metric of connectivity in a graph would be the number of

connected components in the graph as a result of the traj ectories of the agents. This measure

has a maj or drawback however, which is easily demonstrated with the following example.

Consider a graph consisting of a nodes. If the number of connected components is 2, then we

are unsure as to how to interpret this. It is possible that the two components contain [n/2] and

[n/2] nodes respectively. On the other hand, it is also possible that the two components contain

1 and n 1 nodes. These are two very different situations that can arise from the connected

components metric, however with an obj ective value of simply 2, we have no intuition of the

resulting network structure.

Another more obj ective measure of connectivity is as follows. Assume that the agents

have omnidirectional antennas and that two agents in the network are connected if the distance

between them is less than some radius r. More specifically, let 6 : V xV H R represent

the Euclidean distance between a pair of nodes in the graph. Then, we can define a function

c : V xVB {0(, 1} such that

c~pt~us)Pt~uy)) if 6(pt(ui),pt(j>)) r ,(61
c~pt~ui>pt~uj>>0, otherwise. 61

With this, we can define the CCPMANET-D as the following optimization problem:

max cp ),t())(6-2)
t=1 n,vEU

s.t. iij(vy_1,y) < C 4 Vp= {vy1,v2a,..., v,}, (6-3)
pl (u) = s, Vu E U, (6-4)

p (U) = d, Vu E U, (6-5)

where constraint (6-3) ensures that the length of each path pi is less than or equal to its

maximum allowed length 4.

The reader is referred to the paper by Oliveira and Pardalos [137] for additional integer

programming formulations in which other obj ectives are considered and discussed. We finish this

section by providing two results related to the computational complexity of the problem.

Theorem 19. Finding an optimal solution for an instance of the COOPERATIVE


This result, due to Oliveira and Pardalos [137], follows by a reduction from MAXIMUM

3-SAT [79]. We now extend this result in the following theorem.

Theorem 20. Consider an instance of the CCPMANET-D, 11 ithr T as the tinte-horizon. Finding an

optimal sohttion at each tinte-step t E [ 1, T ] is HiP-hard.

Proof: We will show this result by reducing CLIQUE to CCPMANET-D at an arbitrary time-step.

Recall that the CLIQUE problem is as follows. Given a graph G = (V, E) and an integer J < |V|,

does G contain a clique, or complete subgraph, of size J or more [79]?

Consider an instance of CCPMANET-D at any time step t. An optimal solution is one in

which all the agents are pairwise connected. Thus, for n agents the number of connections in an

optimal solution is n(n 1)/2. Notice that this is equivalent to finding clique on n nodes of the

graph. Therefore, given an instance of CLIQUE, by letting J = n, we have the result. Thus there

is a bij section between optimal configurations of agents and cliques in the graph. O

Corrolary 1. For any instance of CCPMANET-D, an upper bound on the optimal sohttion is given

u(u 1)
T (6-6)

where T is the time horizon and u = |U | is the number of agents.

Proof: This proof follows directly from Theorem 20. If all u agents communicate at a given time,

then they form a clique on u nodes. The clique will contain u(u 1)/2 vertices representing the

communication links. If the agents maintain the clique formation over all time steps, then the

number of communication connections will be

u(u 1)
T -(6-7)

and the lemma is proved. O

6.3 Algorithms for CCPMANET-D

6.3.1 Construction Heuristic

In this section, we propose a construction heuristic to quickly create high quality solutions

for the CCPMANET-D. Our obj ective is to provide a fast way of constructing a set of paths,

connecting wireless agents from their initial positions S to the destinations D such that the

resulting routes are feasible for the problem. The union of such sequences of nodes will uniquely

determine the cost of the solution, which is calculated using equation (6-2). The algorithm also

tries to create solutions that have as large a value as possible for the obj ective function.

The pseudo-code for the construction heuristic is showed in Figure 6-1. The algorithm starts

initializing the cost of the solution to zero. The incumbent solution, represented by the variable

solution, is initialized with the empty set.

The next step consists of finding shortest paths connecting each source as E S to a

destination di E D. Standard minimum cost flow algorithms can be used to calculate these

shortest paths. For example, the Floyd-Warshall algorithm [77, 170] can be used to compute the

shortest path between all pairs of nodes in a graph. The Dijkstra algorithm [59] can also be used

to perform this step of the algorithm (with the only difference that, being a single-source shortest

path algorithm, it must be run for | U| iterations, one for each of the | U| source-destination pairs).

In the loop from lines 4 to 10, the algorithm performs the assignment of new paths to the

solution, using the shortest path algorithm described above. First, a source-destination path as-di

is selected, and based on this a shortest path pi corresponding to this pair is generated. Notice

that, if the length (number of edges) of the shortest path pi is more than T there is not feasible

solution for the problem, since the destinations cannot be reached at the end of the requested time

horizon. The algorithm checks for this condition on line 6.

If all source-destination pairs are found to be feasible, then a solution is generated by the

union of all pi. Notice that once agent i reaches node di it can simply loiter at di during all

remaining time (until instant T), as shown in line 7. The sequence of nodes found as a result of

this process is then added to the solution in line 8 of the algorithm, and the optimum obj ective

procedure ShortestPath(G, U, S, D, T)
S c <-
2 solution <- Q)
3 Compute all shortest paths SP(si, di) for each pair (s d ) E Sx D
4 for i = 1 to | U| do
5 ~Pi --SP (si, di)
6 if length of pi > T then
7 return Q)
8 else
9 ~solution <- solution U ~Pi
to c <- c + new connections generated by pi
11 end
L2 end
13 return (c, solution)
end procedure ShortestPath

Figure 6-1: Pseudo-code for the shortest-path construction heuristic.

value is updated (line 9). Finally, a complete solution is returned on line 11, along with the value

of that solution.

Theorem 21. The construction algorithm presented above finds a feasible solution for the

CCPMANET-D in O(|V|3) ie

Proof: A feasible solution for this problem is given by a sequence of positions starting at si and

ending at di, for each agent ui E U. Clearly, the union of the shortest paths provide the required

connection between each source-destination pair, according to the remarks in the preceding

paragraph; therefore the solution is feasible. Suppose that, in line 3, we use the Floyd-Warshall

algorithm for all-pairs shortest path [77, 170]. This algorithm runs in O(|V|3) time. Then, at

each step of the for loop we need only to refer to the solution calculated by the Floyd-Warshall

algorithm and add it to the variable solution. This can be done in time O(|V|), and therefore

the f or loop will run in at most O(| VI ||U|) time. Thus the step with highest time complexity

is the one appearing in line 3, which implies that the total complexity of the algorithm is

o(|v"> V|

6.3.2 Local Improvement

A construction algorithm is a good starting point in the process of solving a combinatorial

optimization problem. However, due to the HiP-hard nature of the CCPMANET-D, such an

algorithm provides no guarantee that a good solution will be found. In fact, it is possible that for

some instances the solution found by the construction heuristic is far from the optimum, and not

even a locally optimal solution.

To guarantee that the solution found is at least locally optimal, we propose a local search

algorithm for the CCPMANET-D. A local search algorithm receives as input a feasible solution

and, given a neighborhood structure for the problem, returns a solution that is guaranteed to be

optimal with respect to that neighborhood.

For the CCPMANET-D, the neighborhood structure is defined as follows. Given an instance

II of the CCPMANET-D, let S be the set of feasible solutions for that instance. Then, if s ES is

feasible for II, the neighborhood Ni(s) of a is the set of all solutions s' E S that differ from s in

exactly one route. Obviously, considering this neighborhood, there are | U| positions where a new

path could be inserted; moreover, the number of feasible paths between any source-destination

pair is exponential.

Thus, in our algorithm, instead of exhaustively searching the entire neighborhood for each

point, we probe only | U| neighbors at each iteration (one for each source-destination pair). Also,

because of the exponential size of the neighborhood, we limit the maximum number of iterations

performed to a constant Maxlte r.

We use randomization to select a new route, given a source-destination pair. This is done in

our proposed implementation using a modified version of the depth-first search algorithm [47].

A randomizeddephrl-fir~st-search is identical to a depth-first search algorithm, but at each step the

node selected to explore is uniformly chosen among the available children of the current node.

Using the randomized depth-first search we are able to find a route that may improve the solution,

while avoiding being trapped at a local optimum after only a few iterations.

procedure HillClimb(solution)
S c <- f (solution)
2 while solution not locally optimal and iter < Maxlter do
3 for i = 1 to | U| do
4 solution <- solution \ {~Pi}
5 Pf<- DFS(si, di)
6 c' <- f(solution U Pf)
7 if length of Pf < T and c > c then
8 c <-C '
9 ~iter <- 0
to else
11 Restore path pi
12 end
13end for
14 iter <-iter+1
15 end while
16 return (solution)
end procedure HillClimb

Figure 6-2: Pseudo-code for the Hill Climbing intensification procedure.

A description of the local search procedure in form of pseudo-code is given in Figure 6-2.

The algorithm used can be described as follows. Initially, the algorithm receives as input the

basic feasible solution generated on phase 1 (the construction phase). A neighborhood for this

solution is then defined to be the set of feasible solutions that differ from the current solution by

one route, as previously described.

Given the basic feasible solution obtained from the construction subroutine, the

neighborhood is explored in the following manner. For each agent ui E U, we reroute the

agent on an alternate feasible path from as to di (lines 3 to 13). Recall that a path pi is feasible

if the total length of this path is less than 4 and the agent reaches its target node by time T.

This alternate path is created on line 5 using a modified depth-first search algorithm[6]. The

modification to the DFS is a randomization which selects the child node uniformly during each

iteration. This procedure is efficient in that it can be implemented in polynomial time, as shown


procedure OnePass(G, U, S, D, T)
S solution <- ShortestPath(G, U, S, D, T)
2 solution <- HillClimb(solution)
3 return (solution)
end procedure OnePass

Figure 6-3: Pseudo-code for the one-pass heuristic.

Theorem 22. The time complexity of the algorithm above is O(kTu2m), where k = Maxlter, T

is the time horizon, a = | U| and m = | E |

Proof: Notice that the most time consuming step of the algorithm is the construction of a new

path (line 5). However, using a randomized depth-first search procedure this can be done in

O(m) time [6]. Each iteration of the while loop (lines 2 to 13) will perform local improvements

in the solution using the re-routing procedure to improve the obj ective function. An upper bound

on the best solution for an instance of this problem is Tu(u -1)/2 (the time horizon multiplied by

maximum number of connections). Each improvement can require at most Maxl ter iterations

to be achieved. Therefore, in the worst case this heuristic will end after O(kTu2m) time. O

6.3.3 One-Pass Heuristic

The two algorithms described in Sections 6.3.1 and 6.3.2 can be combined into a single

one-pass heuristic for the CCPMANET-D [42]. The pseudo-code for the complete algorithm used

can be seen in Figure 6-3. The new algorithm now behaves as a single-start, diversification and

intensification heuristic for the CCPMANET-D.

The total time complexity of this heuristic can be determined from Theorems 21 and 22.

Taking the maximum of the two time complexities determined previously, we have a total time

of O(max~n3, kTu2m}), where T is the time horizon, a = |U|, a = |V|, m = |E|, and

k = Maxlter is the maximum number of iterations allowed on the local search phase.

The algorithm proposed above was tested to verify the quality of the solutions produced, as

well as the efficiency of the resulting method. The test instances employed in the experiments

were composed of 60 random unit graphs, distributed into groups of 20, each group having

graphs with 50, 75, and 100 nodes. The communication radius of the wireless agents was allowed

Table 6-1: Comparative results between shortest path solutions and heuristic solutions.
Instance Nodes Radius Agents OnePass SP Soln Agents OnePass SP Soln Agents OnePass SP Soln
1 50 20 10 63.6 52.4 15 152 120.8 25 414.66 353.6
2 50 30 10 83.8 58.4 15 182.2 124.4 25 516.2 415.6
3 50 40 10 95.4 67.4 15 228.6 171.8 25 695 474.8
4 50 50 10 115.4 64.4 15 275.8 167.4 25 797.4 I
5 75 20 10 76.8 59 20 270.2 228.6 30 575.2 I.
6 75 30 10 85.8 56 20 299.6 241.2 30 725.4 554
7 75 40 10 96.4 64.4 20 386 261 30 862.6 595.4
8 75 50 10 125 67.8 20 403.2 246.8 30 1082.4 670.8
9 100 20 15 113.6 100.4 25 333.4 269.4 50 1523.2 1258.8
10 100 30 15 166.2 124.4 25 511.2 365 50 1901.4 1515.8
11 100 40 15 203.4 141 25 600.6 389.8 50 2539.2 1749.4
12 100 50 15 255.8 151.8 25 756.8 479.6 50 3271.2 2050. 6

to vary from 20 to 50 units. This has provided us with a greater base for comparison, resulting in

random graphs and wireless units that more closely resemble real-world instances.

The graphs used in the experiment were created with the algorithm proposed by Butenko et.

al [39, 40] in the context of the TDMA MESSAGE SCHEDULING PROBLEM. The routines were

coded in FORTRAN. Random numbers were generated using Schrage's algorithm [166]. In all

experiments, the random number generator was started with the seed value 270001.

Results obtained in our preliminary experiments are reported in Table 6-1. In this table, the

results of the one-pass algorithm (OnePass column) are compared to a simple routing scheme

where only the construction phase is explored (the SP Soln column). The solutions shown in the

table represent the average of the obj ective function values from the 5 instances in each class.

The numerical results provided in the table demonstrate the effectiveness of the proposed

heuristic when the improvement phase is added to the procedure. The proposed heuristic

increased the obj ective value of the shortest path solutions by an average of 3 One reason for

this is the fact that, when agents are routed solely according to a shortest path, they are not taking

advantage of the remaining time they are allotted (i.e. the time horizon T) and the values from

the distance limit given by L.

procedure GRASP(Maxlter, RandomSeed)
S f <-o
2 X*t <--
3for i = 1 to Maxlter do
4 X <- ConstructionSolution (G, g, X)
5 X <- LocalSearch(X, NV(X))
6 if fX) > fX*) then
7 X <-X
s f f (X)
10 end
11 return X*
end procedure GRASP

Figure 6-4: GRASP for maximization

Our heuristic, on the other hand, allows wireless agents to take full advantage of these

bounds. The algorithm can do this by adjusting the paths to include those nodes within the

range of other agents. In addition, at any given time an agent is allowed to loiter in its current

position, possibly waiting for other agents to come into its range. This cannot occur in the

phase 1 algorithm because, according to the shortest path routing protocol, loitering is forbidden.

We notice that our method provides solutions that are better than the shortest path

protocol. The time spent on the algorithm has always been less than a few seconds, therefore

the computational time is small enough for the problem sizes explored in our experiments.

We believe, however, that the quality of the solutions and computational time can be further

improved using a better implementation, and more sophisticated data structures to handle the

information stored during the algorithm.

6.3.4 Greedy Randomized Adaptive Search

In this section, we describe the implementation of the Greedy Randomized Adaptive Search

Procedure (GRASP) (Section 2.7.2) for the CCPMANET-D. Pseudo-code for the generic GRASP

is provided in Figure 6-4. We discuss in this section how the above algorithm can be specialized

to provide approximate solutions for the CCPMANET-D. In the following subsection, we describe

an algorithm for the GRASP construction phase that provides initial solutions for instances of the

CCPMANET-D problem. Then we provide a local search algorithm for the improvement phase.

Construction Phase. The first task in a GRASP algorithm is to build good feasible solutions

in terms of a given obj ective function. To do this, we need to specify the set A, the greedy

function g, the parameter a~, and the neighborhood NV(X), for X E F. The components of

each solution X are feasible moves of a member of the ad-hoc network from a node v to a node

we N (v) U {v}. We say that for an agent ui E U located at node v in the graph that pitv)

represents a shortest path from the current node v to the destination for agent ui, namely node di.

The complete solution is constructed according to the following procedure outlined in the

pseudo-code reported in Figure 6-5. In the figure, ah refers to the current location of an agent.

First, the solution which is initially empty is augmented to include the starting locations for all

agents. Then, the time variable t is initialized to 1, and in line 6 an agent ui E U is selected at

random and routed from its along a shortest path pi(si) from its source node si to its destination

node di. If the total distance of pi(as) is greater than Li, then the instance is clearly infeasible and

the algorithm ends. Otherwise the procedure continues and the remaining agents are scheduled

in the loop beginning at line 8. The procedure considers each feasible move (q, w, u) before

scheduling an agent. A feasible move connects the final node q of a sub-path P,, for a E U, to

another node w, such that the shortest path from w to d, has distance at most L, CE gL dist(e).

The set of all feasible moves in a solution is defined as A(X).

The loop from lines 12-14 ensure that a node currently at its destination remains there.

Likewise, the loop from lines 15-17 schedule an agent & on a shortest path ph(ah) from its

current position ah to dh if the maximum allowed travel time for agent & is equal to the |~Ph(ah)|i.

From 19-21, the set L C A(X) is formed and consists of all feasible moves for agents not yet

scheduled. Then in line 25, the greedy function g returns for each move k E L the number of

additional connections created by that move. As described above, the construction procedure

will rank the elements of L according to g. The best a~k elements are then added to the RCL and

procedure ConstructionSolution (G, g: X)
xi X 8
2 for i = 0 to k~ do
3 X+ X U{ }.
4 end

6 select randomly ui EUI and route usi on shortest-path P (a )
7 X+- X U {P;(si) }
8 while t < T' AND 3 us EUI \ X do

lo RCLT +- 8
rl for ush E Ui\ Xdo
3 2 if a;, = d;,do

142 end
15 if diSt(a;,, d;,) L,-' 4ds~)d
16i route Uh On shortest path Pal(ah)
17 X~ +- X U {p(ah)}
rs end
1g while 3 (1,, 12,, us1) do
20 { l ,1,,
22 end
23 a+- rand(0, 1)
24 6 (Maxnn;Contribute~bc l (aK J (Max~z~l ontribute~bc l Ai2rn.Cont~riibue)))
25 for all (14, lyi, ne) such that g((14, lj ne)) > 6 do
26 R~C:L < -- RCL U { (14,; 6,1L,u)
27 end
28 (14,t lj, ne) <- get randomly from RCL
29 add (1 lj ne) into the path of agent we, in the solution
30 t <- 1
32 return(XY)
end procedure ConstructionSolution

Figure 6-5: Greedy randomized constructor for CCPMANET-D.

in lines 28-29 a move is selected at random and added to the solution. Thi s is repeated until a

complete solution for the problem is obtained.

Improvement phase. In the local search phase, GRASP attempts to improve the solution

built in the construction phase. As mentioned above, we use a hill-climbing procedure where the

procedure LocalSearch(X, MaxlterLS)
S X' <--
2 t <-
3 Lastlmp~rove <- 1
4 i <-2
5 iter <- 0
6 while i / Lastlmp~rove and iter < MaxlterLS do
7 Remove current path from si to di for agent ui
8 while ai / di do
9 if dist(ai, di) Li e ,Pz diSt(e) then
to Route user ui using its shortest path pi(as)
11 else
L2 Bestl~ove <- {(ai, Is,i us)|((as, i,, ui)) > g((ai, ly, ui)), V (ai, ly, ui) }
13 end
14 Add Bestl~ove into the new path for ui in solution X'
15i t<-t + 1
16 end
17 if f(X') > f (X) then
18 X <- X'
19 Lastlmp~rove <- i
20 end
21 i <- i + 1 mod k
22 iter <- iter + 1
23 end
24 return(X')
end procedure LocalSearch

Figure 6-6: Local search for CCPMANET-D.

obj ective is to improve the solution as much as possible until a local optimal solution is found as

described in the pseudo-code provided in Figure 6-6.

The local search receives the construction phase solution X and a parameter MaxlterLS

as input. In each iteration, the neighborhood NV(X) of X is explored in search of a solution X'

such that f (X') > f (X). In order to explore NV(X), a perturbation function is defined as follows.

In the loop in lines 5-21, agents are re-routed using a greedy method similar to that of the

construction phase. In line 6, the current construction phase path for agent ui is removed from the

solution. Then each feasible move is considered and the move which adds the greatest increase

to the obj ective function, Bestl~ove, is added to the new path for agent ui. This is repeated for

all agents until a new feasible solution X' E NV(X) C A(X) is created. If f (X') > f (X), then

in line 17, X' is set as the new current solution. The process returns to line 5 and repeats until no

agent can be re-routed according to this greedy method and improve the current solution or until

some maximum number of iterations MaxlterLS are completed.

6.3.5 Complexity of the Heuristic

The following theorems address the computational complexity of the proposed algorithm.

Theorem 23. 7he construction phase finds a feasible solution for the CCPMANET-D in O(Tmu2)

time, where T is the time horizon, a = |U|, and m = |E(G) |.

Proof: Notice that the while loop from lines 8-31 will require T(|U| 1) iterations to

complete. Likewise, the loop from lines 11-22 requires | U| iterations. Within the loop, the

most time consuming step is the construction of shortest path. However, this can be done using a

breadth-first search in O(m) time [6]. Thus we have the result. O

Theorem 24. 7he time complexity of the local search phase is O(kTu2m), where T is the time

horizon, a = |U|, m = |E(G) |, and k = M~axlterLS.

Proof: The proof is similar to Theorem 23. Notice that the while loop from lines 5-22 perform

local improvements according the greedy re-routing scheme. Again the most time consuming

step is the construction of a shortest path which can be accomplished in O(m) time. Each

improvement can require up to k iterations of the loop. Thus we have the proof. O

Corrolary 2. 7he overall time complexity of the proposed GRASP is O(ITu2m(k + 1) ), where T

is the time horizon, a = |U|, m = |E (G) |, k = M~axlterLS, and I = M~axlter is the overall

number of GRASP iterations.

Proof: The proof is immediate from Theorem 23 and Theorem 24. O

Path-relinking .

First introduced by Glover in [84], path-relinking (PR) was used as an enhancement for

tabu search heuristics. PR was first combined with GRASP by Laguna and Marti [125]. When

applied to GRASP, path-relinking introduces a memory to the heuristic which usually results

in improvements in solution quality. This is because in the standard GRASP framework, the

procedure PathRelinking(x,, 8)
S x, <- randSelect (y E : A(x,, y) > 6)
2 f max {f (x,), f (x,) }
3 x* <- arg max {f (x,), f (x, )

5 while A(x,, x,) / 0 do
6 m* <- arg max f (x 8 m) : m E a(x, x,)}

9 if f (x) > f then
1o f f (x)
11 Z 4--
12 end
14 return x*
end procedure PathRelinking

Figure 6-7: Path-relinking subroutine.

multi-start nature of the heuristic does not include any long-term memory mechanism for

saving traits of good solutions generated by the algorithm. Path-relinking allows GRASP to

remember these traits and favor them in successive iterations. GRASP with path-relinking has

been successfully applied to problems such as MAXIMUM CUT [71], QUADRATIC ASSIGNMENT


[125]. For a survey of GRASP with path-relinking, the reader is referred to [158].

Path-relinking works by maintaining a set of elite solutions 8, known as guides and

examines point-to-point traj ectories between a guiding solution and an incumbent solution

in search of an optimum. Pseudo-code for a generic path-relinking procedure is provided

in Figure 6-7. To perform path-relinking, we begin with a guiding solution x, E 8, and an

initial starting solution x,. The guiding solution x, is selected at random from the pool of elite

solutions 8, so long as the symmetric difference A(x,, x, ) between the two solutions x, and

x, is sufficiently large. The symmetric difference is defined as the set of pairwise exchanges

needed to transform x, in to x,. Recall that all solutions in 8 are local optima, and we are trying

to discover solutions which are not located in the neighborhoods of x, or x,. Therefore this

constraint prevents us from applying path-relinking to solutions which are too similar to each

other, and would not likely yield an improved solution [70].

At each step, the procedure examines all moves me A(x, x,), and greedily selects the move

which results in the maximum increase in the obj ective of the current solution. This occurs in line

6 of the pseudo-code in which the move m* is selected as the move which maximizes f (x e m),

where x e m is the solution which results from incorporating m in to x. In line 7, the symmetric

difference is updated, and if necessary the best solution is updated in lines 9-12. The procedure

ends when A(x, x,) = Q), i.e. when x = x, [158].

Path-relinking can be applied to a pure GRASP in a straightforward manner, which can be

visualized in the pseudo-code of Figure 6-8. First, the set of elite solutions 8 is initialized to the

empty set in line 2 and is built by including the solutions from the first MaxElite iterations. After

a standard GRASP iteration of greedy randomized construction and local search produces a local

optimal solution X, the PathRelinking procedure is called on line 7. For the CCPMANET-D,

the elements in the symmetric difference are the agent paths which differ between the initial and

guiding solutions. The value of m* from Figure 6-7 is the path for an agent in the symmetric

difference which results in the maximum increase in the total number of communications

between the agents. In line 8, a function UpdateElite is called in which the elite pool is

possibly updated. The solution returned from path-relinking is included in the elite pool if it

is better than the best solution in 8 or if it worse than the best but better than the worst and is

sufficiently different from all elite solutions [158]. Finally, the optimal solution is updated in

lines 12 to 14 if necessary.

6.3.6 Computational Experiments

The proposed procedure was implemented in the C programming language and compiled

using the MicrosoftO Vi sual C++ 6.0. It was tested on a PC equipped with a 1 800MHz Intel ~

PentiumB 4 processor and 256 megabytes of RAM operating under the MicrosoftB WindowsB

2000 Professional environment.

procedure GRASP+PR(Maxlter, RandomSeed)

3for i = 1 to Maxlter do
4 X <- ConstructionSolution (G, g, X)
5 X <- LocalSearch(X, MaxlterLS)
6 if |E| = MaxElite then
7 X <- PathRelinking(X, 8)
8 UpdateElite (X, 8)
to 8 <--SU {X
11 end
L2 if f (X) > f (X*) then
13 X* <-X
14 end
15 end
16 return X*
end procedure GRASP+PR

Figure 6-8: GRASP with path-relinking for maximization.

Both the pure GRASP and the GRASP with path-relinking were tested on a set of 60

random unit graphs with varying densities 20 each having 50, 75, and 100 nodes. The radius of

communication varies from 1 to 5 units (miles) in unit increments. We tested each case with three

sets of mobile agents to achieve better comparisons and model real-world scenarios. Thus, in

total 900 test cases were examined. The graphs were created by a generator used by Butenko et


Since any instance of the CCPMANET-D is composed of several parameters, i.e. the number

of mobile agents, their respective source and destination nodes, the radius of communication,

and the maximum time horizon, each of which impacts the optimal solution for the instance, we

will provide our numerical results in several sets of tables. First, we report solutions for several

representative instances and provide all input parameters in order to establish an inference base

for the overall experiment. Then we will summarize the overall results by providing the average

solutions for each problem set.

Table 6-2: Three instances with different sets of agents on 50 node graphs are given. The value in
the UBound column was found using Corollary 1.
Instance: 50r30il Nodes: 50 Agents: 10 MaxTime: 10
Source: [ 6 10 10 3 5 7 4 2 10 6 ]
Destination: [ 49 47 44 48 46 40 48 42 47 47 ]
1 291 303 450
2 365 373 450
3 412 423 450
4 443 443 450
5 449 449 450

Instance: 50r30il Nodes: 50 Agents: 15 MaxTime: 10
Source: [ 109 89 68 17 25 55 2 11]
Destination: [ 49 47 44 48 46 40 48 42 47 47 ]
1 756 757 1050
2 881 909 1050
3 963 972 1050
4 1029 1029 1050
5 1050 1050 1050

Instance: 50r40i4 Nodes: 50 Agents: 25 MaxTime: 10
Source: [89845446274217589381187568]
Destination: [ 49 48 44 48 46 42 49 40 48 49 45 46 49 45 48 44 42 41 48 43 40 49 45 49 43 ]
1 2613 2653 3000
2 2896 2918 3000
3 3000 3000 3000
4 3000 3000 3000
5 3000 3000 3000

In Table 6-2, we report solutions for three different instances on 50 node graphs. The

Source vector and Destination vector provide the respective (as, di) pair for each agent

respectively. The specific values of si were randomly selected from the first 21 1' of the nodes

of the graph. Likewise, the di values were chosen randomly from the last 21 1' of nodes. This

method of selection is preferred over a completely randomized design because in real-world

situations such as a combat scenario, the available entry and exit points from a battle space are

likely to be limited. However, using a random selection from the available subset of nodes allows

for more thorough testing and helps avoid unintentional biases.

Table 6-3: Three instances with different sets of agents on 75 node graphs are given. The value in
the UBound column was found using Corollary 1.
Instance: 75r30i2 Nodes: 75 Agents: 10 MaxTime: 15
Source: [ 7 6 13 3 10 13 15 6 6 2 ]
Destination: [ 68 68 71 73 68 68 73 70 74 62 ]
1 571 575 675
2 614 621 675
3 658 658 675
4 670 670 675
5 675 675 675

Instance: 75r40i4 Nodes: 75 Agents: 20 MaxTime: 15
Source: [ 115 7153 89 4 613 143 8 31014 1115 9 3]
Destination: [ 63 66 69 61 62 68 62 67 68 66 62 60 61 66 63 73 72 64 71 71 ]
1 2535 2554 2850
2 2746 2758 2850
3 2842 2842 2850
4 2850 2850 2850
5 2850 2850 2850

Instance: 75r30il Nodes: 75 Agents: 30 MaxTime: 15
source: [141528104133451204023158512037711341154]
Destination: [ 08 00 03 02 05 72 02 02 o~ 71 05 00 03 04 00 04 00 00 00 04 74 03 73 04 04 03 05 05 00 03 ]
1 4721 4870 6525
2 6002 6012 6525
3 6265 6285 6525
4 6497 6497 6525
5 6525 6525 6525

The column MaxTime is the maximum time horizon T. Recall that all agents must reach

their destination node by this time. The GRASP column provides the solution from GRASP after

1000 iterations and UBound is the upper bound on the solution value and was calculated by the

equation in Corollary 1. Notice that as the radius value increases, the number of connections

between the agents tends to converge to the value of the upper bound. Recall that the upper

bound value from Corollary 1 is not an upper bound on the optimal solution for the given graph

per se; it is an upper bound on the solution for the given time horizon and number of agents.

Thus, the more dense the graph, the tighter the bound.

Table 6-4: A 100 node instance with solutions with radius varying from 1 to 5 units. The value in
UBound was found using Corollary 1.
Instance: 100r30i2 Nodes: 100 Agents: 15 MaxTime: 20
Source: [ 9 19 10 18 13 18 12 18 15 8 6 6 20 18 1 ]
Destination: [ 84 88 83 84 96 96 81 95 83 82 93 80 90 85 81 ]
1 1819 1821 2100
2 1960 1974 2100
3 2065 2067 2100
4 2100 2100 2100
5 2100 2100 2100

Instance: 100r30il Nodes: 100 Agents: 25 MaxTime: 20
Source: [ 17 6 9 19 9 12 2 15 7 8 1 2 8 6 3 13 16 17 13 13 17 19 2 5 21 ]
Destination: [ 81 89 84 82 88 99 93 89 93 97 84 96 96 91 90 86 98 86 81 89 82 89 81 80 99 ]
1 5183 5186 6000
2 5577 5647 6000
3 5898 5909 6000
4 5992 5992 6000
5 6000 6000 6000


Instance: 100r30i2 Nodes: 100 Agents: 35 MaxTime: 20
source: [35112144451012141317417810~17151312 05 011308002
Destination: [ so as so at 84 00 as at ea sa as so as so as as 02 so 81 85 as 04 sa so oo 0 061 oa co 0 00 00 00 st sa 82
1 10222 10255 11900
2 11108 11224 11900
3 11660 11704 11900
4 11842 11845 11900
5 11900 11900 11900

Table 6-3 presents the specific parameters and related solutions for three instances of the

CCPMANET-D on 75 node graphs. On these networks, the number of agents varied from 10 to

30, and the maximum time horizon was 15. Again, we see that as the communication radius

increases the solutions tend to the upper bound values. Similar results for three graphs having 100

nodes are provided in Table 6-4. For the 100 node instances, the number of agents varied from 15

to 35 and the maximum travel time was 20 units. The results for these instances also indicate that

the heuristic is robust and able to provide excellent solutions for large instances.

Table 6-5: Average solution values for GRASP and GRASP with path-relinking on 50 node
Nodes Agents Radius GRASP GRASP+PR Bound
50 10 1 347 352.21 450
50 10 2 404.58 407.58 450
50 10 3 428.32 429.47 450
50 10 4 437.84 438.53 450
50 10 5 444.37 444.58 450
50 15 1 813.11 817.32 1050
50 15 2 937.74 945.47 1050
50 15 3 1001.11 1003.58 1050
50 15 4 1025.37 1026.21 1050
50 15 5 1037.16 1037.53 1050
50 25 1 2272.79 2315.58 3000
50 25 2 2686.26 2704.53 3000
50 25 3 2850.84 2861.95 3000
50 25 4 2924.05 2927.68 3000
50 25 5 2959 2959.26 3000
Average Comp Time (s) 2.89 4.29

Table 6-6: Comparative solutions of GRASP and GRASP with path-relinking on 75 node graphs.
Nodes Agents Radius GRASP GRASP+PR Bound
75 10 1 574.95 577.42 675
75 10 2 629.42 631.37 675
75 10 3 653.53 654.63 675
75 10 4 665.42 665.89 675
75 10 5 669.47 669.84 675
75 20 1 2288 2319.63 2850
75 20 2 2639.37 2651.5 2850
75 20 3 2756.69 2762 2850
75 20 4 2805.53 2807.68 2850
75 20 5 2827.42 2828.42 2850
75 30 1 5349.84 5391.26 6525
75 30 2 6037.47 6064 6525
75 30 3 6310.90 6332.37 6525
75 30 4 6422.11 6430.80 6525
75 30 5 6472.42 6478.84 6525
Average Comp Time (s) 6.16 7.43

Tables 6-5, 6-6, and 6-7 show the evolution of the average solution values as the

communication range increases for the 50, 75, and 100 node graphs, respectively. Notice

once more that as the communication range increases, the average solution converges to the value

Table 6-7: Results of GRASP and GRASP with path-relinking on 100 node graphs.
Nodes Agents Radius GRASP GRASP+PR Bound
100 15 1 1838.25 1840.45 2100
100 15 2 1996.75 2003.15 2100
100 15 3 2061.9 2064.7 2100
100 15 4 2083.1 2084.4 2100
100 15 5 2093.95 2094.05 2100
100 25 1 4979.1 5019.2 6000
100 25 2 5655.3 5674.35 6000
100 25 3 5869.35 5876.9 6000
100 25 4 5940.65 5944.7 6000
100 25 5 5978.2 5979.2 6000
100 35 1 9947.45 9997.15 11900
100 35 2 11254.55 11280 11900
100 35 3 11636.85 11664.5 11900
100 35 4 11787.9 11793 11900
100 35 5 11859.1 11860.35 11900
Average Comp Time (s) 5.17 8.05

of the upper bound given by Corollary 1. In these tables we also report the average computing

time required by both the pure GRASP and the GRASP+PR to find their best solutions within

the specified number of iterations. For all of the experiments, the GRASP+PR found solutions at

least as good as the pure GRASP, finding superior solutions for 45' of the instances tested.

In Figures 6-9, 6-10, and 6-11, we provide plots of the average objective function value

versus communication range found using GRASP with path-relinking. The upper bound values

for each case as computed by Corollary 1 are also plotted in the charts. These graphs indicate that

on average, as the radius of communication increases, the obj ective function values tend to the

upper bound values.

6.4 A Continuous Formulation (CCPMANET-C)

In this section, we present a continuous formulation for the CCPMANET-D. This formulation

will provide a more realistic scenario than the discrete formulation provided above. We will

assume that the agents are operating in battlespace QC cRd, where Q is a compact, convex set

with unit volume and the Euclidean norm |I | |~ |2 d. For our purposes, we are going to consider

the planar case, i.e. d = 2, with the understanding that extensions to higher dimensions are

300025 agents upper bound


cn 2000

0 1500

15 agentsupebon

500- 10aet pebod
1 2 3 4 5
Communication radius

Figure 6-9: Evolution of GRASP+PR solution values on 50 node graphs as the communication
radius increases from 1 to 5 units.

possible. Suppose there are NV wireless agents in the ad hoc network. The NV agents are assumed

to be omnidirectional and are modeled as point masses. We will suppose that the agents are free

to move within Q at some bounded velocity. Assume without the loss of generality, that the

maximum velocity magnitude is unitary, i.e. | | v (t)l | | < for ie { 1, .. ., N}).

In order to derive a continuous formulation, we need to to define an obj ective function that

is consistent with that of the discrete formulation. Let Ri be the communication constant for

agents i and j. That is, Rij is the radius of communication for the two agents. Then one possible

obj ective is a so-called heavyside function defined as

0 if || ,t ~,||2 > ij* 68

A graphical representation of H1 is given in Figure 6-12. While this function will work as an

obj ective, it is very extreme. That is, if H [Ri |4 IIr~ 2 ~ ] = 0, then there is no information

30 agents upper bound

6000 C;- SP+PR



300 20agents upper on


10 agents __ ________ pper bond
1 2 3 4 5
Communication radius

Figure 6-10: Evolution of GRASP+PR solution values on 75 node graphs as the communication
radius increases from 1 to 5 units.

provided which might indicate where a better solution might lie. A more desirable function

would be one that approximates H1 but is continuous.

We consider to alternatives to H1. The first function is a piecewise continuous function

given as

H2[, S33a r~ r ific c R ,~< ||r-r 2 < R (6-9)

0, if || ,r X4,||2 > 2Rij.

This function whose graph is provided in Figure 6-13 has a value equal to one if agents i and j

are within the communication radius Rij of one another. The function then decreases constantly

until the agents are 2Rij apart at which time they are unable to communicate.

35 agents upper bound




a 00 25 agents upper bound


15 agents upper bound
::- --- IRASP+PR

1 2 3 4 5
Communication radius

Figure 6-11: Evolution of GRASP+PR solution values on 100 node graphs as the communication
radius increases from 1 to 5 units.

The third and final obj ective function we will consider is a continuously differentiable

decreasing function of the distance between agents i and j and the communication radius Rij .

This function, given as

H3 r r2,i (6-10)

can be seen in Figure 6-14. This is perhaps the best approximation of H1 in that it can be

interpreted as the probability of agents i and j communicating when they are some distance

Now that we have found a suitable obj ective function we can define the remaining

parameters and constraints of the problem. Let x" (t) be the position of agent i at time

t. Then, we can describe the location of all NV agents at a time t E Z+ as an N~-vector

X(t) = (xl(t), x2() ~ Nt) E N. Similarly, let v(t) be the velocity of agent i at

time t. The relationship between velocity and position is typical and is given by (~t)=.

SI 15


Figure 6-12: The heavyside function H1.

In order to formulate the continuous time analog of the CCPMANET-D, we must constrain the

maximum velocity of each agent. This will enforce the constraints on the maximum distance

traveled in the discrete formulation. If Si E IR2 is the starting position of agent i, and Di E IR2
is the destination point of agent i, then we can formulate the CONTINUOUS COOPERATIVE


Maximize H3CI, 2~i'l ,ij]


x (0) = Si,

F (T) = Di,

if(t) = %,

x"i(t) E IR2,

1, 2, ... N, tE [0, T] ,

=, 12,...,NV, tE [0,T].





tr tr

1li 15 Fl

Figure 6-13: Alternate objective function H2-

Using this formulation as a starting point, heuristics for continuous global optimization

problems can be implemented. Currently we are developing an algorithm based on the

Continuous Greedy Randomized Adaptive Search Procedure (C-GRASP) proposed by Hirsch

et al. [101]. Subsequent work with C-GRASP including enhancements and stopping rules can

be found in [100]. C-GRASP has also been used to solve systems of nonlinear equations [99],

and for solving continuous formulations of discrete optimization problems [38]. This work is

currently in process and the results will appear in a paper later this year [41].

6.5 Concluding Remarks


AD HOC NETWORKS. We presented both discrete and continuous formulations, discussed the

computational complexity, and presented several algorithms for solving each formulation.

Furthermore, extensive computational results were presented which show the effectiveness of

the proposed algorithms. Lastly, we derived a continuous formulation of the problem. Using

Figure 6-14: Second alternate objective function H3

this version, several heuristics for continuous optimization problems can be applied to achieve

solution which more closely mirror real-world situations. This is because we have removed the

underlying graph structure and the motion of the agents is constrained kinematically.

Future work on the problem of path-planning for a group of wireless users such as the

one presented here should could focus on a multiobj ective problem in which the agents not

only maximize the communication time but also maximize the amount of the battlespace (i.e.

the communication graph) that is covered. This formulation would be particularly useful in

combat search and rescue operations and other reconnaissance applications. In the formulation

considered, it is still possible that the agents will loiter at a given node until such time passes that

they must proceed to their destinations. Instead of merely circling a single node, the obj ective of

maximum battlespace coverage will still allow the agents to maximize the communication, while

visiting many areas of the region.

1/5rl-;u brj I

In the following IIh liphi r we take a closer look at the actual mechanisms used for

communication in the MANET described in this chapter. Instead of the generalized view of

agent connectivity used here, we examine a particular type of transceiver which may be used

by the agents. In particular, we consider the time division II1 r thiple access (TDMLA) style radios.

TDMA transceivers are popular because they make efficient use of the available bandwidth by

allowiing frequency reuse. Not surprisingly however, it turns out that several problems must be

mitigated in order to ensure effective group communication.


7.1 Introduction

The MANET such as the one described in the previous chapter is an example of a so-called

wireless mesh network (WMN). WMNs have become an important means of communication in

recent years. In these networks, a shared radio channel is used in conjunction with some packet

switching protocol to provide high-speed communication between many users. The stations

in the network act as transmitters and receivers, and are thus capable of utilizing a multi-hop

transmission procedure. The advantage of this is that several stations can be used as relays to

forward messages to the intended recipient, allowing for beyond line of sight communication for

stations that are geographically disbursed and potentially mobile [39].

Mesh networks have increased in popularity in the recent years and the number of

applications is steadily increasing [155]. As mentioned in [8], WMNs allow users to integrate

various networks, such as Wi-Fi, the internet and cellular systems. WMNs can also be utilized in

a military setting in which tactical datalinks network various communication, intelligence, and

weapon systems allowing for streamlined communication between several different entities. For a

survey of wireless mesh networks, the reader is referred to [8].

In WMNs, the critical problem involves efficiently utilizing the available bandwidth to

provide collision free message transmissions. Unfettered transmission by the network stations

over the shared channel will lead to message collisions. Therefore, some medium access control

(MAC) scheme should be employed to schedule message transmissions so that collisions are

prevented. The time division multiple access (TDMA) protocol is a MAC scheme introduced

by Kleinrock in 1987 which was shown to provide collision free broadcast schedules [117]. In

a TDMA network, time is divided into frames with each frame consisting of a number of unit

length slots in which the messages are scheduled. Stations scheduled in the same slot broadcast

simultaneously. Thus, the goal is to schedule as many stations as possible in the same slot so long

as there are no message collisions.

When considering the message scheduling problem on TDMA networks, there are

two optimization problems which must be addressed [175]. The first involves finding the

minimum frame length, or the number of slots required to schedule all stations at least once. The

second problem is that of maximizing the number of stations scheduled within each slot, thus

maximizing the throughput. In this chapter, we consider the MESSAGE SCHEDULING PROBLEM

ON TDMA NETWORKS (MSP-TDMA). We provide a integer programming formulation and

prove that the problem is HiP-hard. We then derive several heuristics and compare their

performance against other algorithms for the literature. Extensive computational results indicate

the superiority of our methods against real-world instances.

7.2 Problem Description

A TDMA network can be conveniently described as a graph G = (V, E) where the vertex

set V represents the stations and the set of edges E represents the set of communication links

between adj acent stations. There are two types of message collisions which must be avoided

when scheduling in TDMA networks. The first, called a direct collision occurs between one-hop

neighboring stations, or those stations i, j E V such that (i, j) E E. One-hop neighbors which

broadcast during the same slot cause a direct collision. Further, if (i, j) 5' E, but (i, k) E E and

(j, k) E E, then i and j are called two-hop neighbors. Two-hop neighbors transmitting in the

same slot cause a hidden collision [39].

Assume that there are M~ slots per frame. Further, assume that packets sent at the beginning

of each time slot and are received in the same slot in which they are sent. Let x : M~x V

{0, 1}, be a function where

1, if station n scheduled in slot m,
Xmn ~0, otherwise. 71

Also, let c : E { 0, 1} return 1 if i and j are one-hop neighbors, i.e., if (i, j) E E and i / j.

As mentioned above, there are two problems which have to be solved in order to obtain

optimal broadcast schedules using the TDMA protocol. The first is the FRAME LENGTH


PROBLEM (TMP). Using the aforementioned definitions and assumptions, we can now formulate


multiobj ective optimization problem:

Minimize M~

Maximize xi;,
i= 1 j= 1
subj ect to

xmn >1, VnE V, (7-2)

ce + xmi xmy
cik mi Cksj mj < 1, Vi ,keV ,j/k ,m=1 (7-4)

Xmn E {0, 1},Vn )=1.. (7-5)

M~EZ (7-6)

The obj ective provides a minimum frame length with maximum bandwidth utilization,

while constraint (7-2) ensures that all stations broadcast at least once. Constraints (7-3) and

(7-4) prevent direct and hidden collisions, respectively. We note here that will not be attempting

to solve this problem by using the typical multiobj ective optimization approach, in which one

combines the multiple obj ectives into one scalar obj ective whose optimal value is a Pareto

optimal solution to the original problem. Instead we will decouple the obj ectives and handle each

independently. This is done because for the MSP-TDMA, frame length minimization usually takes

precedence over the utilization maximization problem. This is the usual modus operandi used by

other heuristics in the literature [164, 169, 175].

Suppose that we relax the MSP-TDMA and only the consider the first obj ective function.

This is referred to as the FRAME LENGTH MINIMIZATION PROBLEM (FLMP) and is given by

the following integer program: min{M~ : (7-2) (7-6)}. Clearly any feasible solution to this

problem is feasible for MSP-TDMA. Now, consider a graph G' = (V, E') where V follows

from the original communication graph G, but whose edge set is give yE U{(,j

i, j are two-hop neighbors}. Then using this augmented graph, we can formulate the following

theorem .

Theorem 25. Thze FRAME LENGTH MINIMIZATION PROBLEM on G = (V, E) is equivalent to

finding an optimal coloring of the vertices of G' (V, E').

Proof: Recall that in order for a message schedule to be feasible, all stations must broadcast

at least once and no collisions occur, either hidden or direct. Notice now that E' contains both

one-hop and two-hop neighbors, and in any feasible solution, neither of these can transmit in the

same slot. Thus, there is a one-to-one correlation between time slots in G and vertex colors in G'.

Hence, a minimum coloring of the vertices of G' provides the minimum required slots needed for

a collision free broadcast schedule on G. O

After one has successfully solved the FLMP yielding an optimal frame length M~*, then

the THROUGHPUT MAXIMZATIO PROBLE (TMP) give as"" follows max,,,,, {pM xy :

(7-2) (7-6) } can be solved, where M~ is replaced by M~* in (7-2) (7-6). It turns out that

both the FLMP and the TMP have been shown to be HiP-hard [39, 67]. Thus it is unlikely that

a polynomial algorithm exists for finding the optimal broadcast schedule [79]. It is interesting

to note however, that if we ignore constraint (7-4) which prevents two-hop neighbors from

transmitting simultaneously, then the resulting problem is in p, and a polynomial time algorithm

is provided in [93].

7.3 Computational Complexity

This section presents the computational complexity results for the MSP-TDMA. It was first

noted that the MSP-TDMA is HiP-complete by Wang and Ansari in [169]. However, their proof

of the HiP-completeness of the recognition version of the problem was incorrect due to some

faulty arguments. Namely, they claimed that the GRAPH COLORING PROBLEM is equivalent

to the MAXIMUM INDEPENDENT SET PROBLEM based on an incorrect assumption that, given

an arbitrary graph, an optimal coloring can be found by recursively computing a maximum

independent set and removing it from the graph. Thus, by coloring different independent sets in

Figure 7-1: Counterexample to the claim of Wang & Ansari that optimal graph coloring can be
found by recursively finding a maximum independent set and removing it from the graph.

different colors, they claim that the chromatic number of the graph equals the total number of

independent sets computed.

Figure 7-1 presents a counterexample to this statement. It is easy to see that the

independence number of the graph in this figure is 3. Assuming that the first maximum

independent set found using the so-claimed "optimal" coloring algorithm of Wang and Ansari

is {4, 5, 6} and consequently removing this set from the graph, we obtain a clique (complete

subgraph) of the three vertices {1, 2, 3}. The independence number of the remaining graph

is 1, so all three of the remaining vertices have to be colored in different colors. Thus, the

Wang-Ansari coloring algorithm results in a 4-coloring. However, it is easy to see that the

chromatic number of this graph is 3. For example, one optimal coloring is given by the following

partition: {1, 5, 6}, {2, 4}, {3}. Therefore, the coloring obtained using the Wang-Ansari approach

is not optimal.

Next we prove that the recognition version of the MSP-TDMA is in fact HiP-complete. We

consider the following problem:


INSTANCE: A undirected graph G = (V, E) and an integer K.

QUESTION: Does there exist a broadcast schedule with frame length < K?

Theorem 26. Thze TDMA MESSAGE SCHEDULING PROBLEM is NiP-complete.

Figure 7-2: Construction of graph G' from G.

Proof: To show that K-MSP is HiP-complete, we need to show that (1) K-MSP E NiP; (2)
Some HiP-complete problem reduces to K-MSP in polynomial time. Suppose that n = | V|

and m = |E|. Without the loss of generality, we assume that G is connected (if it is not, we can

consider each connected component separately).

K-MSP E NiP since a given broadcast schedule with frame length k < K can be verified for

validity in O(n3) time. Indeed, the verification of validity consists of checking, for each vertex

i E V, that the set Li of all time slots in which the vertices from {i} [] N(i) transmit according
to the given schedule does not contain any repeated elements. This can be done using the sorting

of time slot numbers in L, in O((|~L| + 1 ) log(|Le|i + 1)) time for vertex i, therefore the total run
time will be

OIi= (|Le| + 1i) loC)= ((mb + n) log(u)) = O(rL3)

We will show that the graph k-coloring problem can be reduced to K-MSP in polynomial

time. Recall that the k-coloring problem is, given G = (V, E) and an integer k, does there exist

a proper coloring of the vertices of G that uses < k colors? This is a well-known NiP-complete

problem [79].

Given a graph G = (V, E), we will construct the corresponding graph G' = (V', E'),

where V'' = vU E~ and E' = { id, (i, j)] : (i, j) E E, i, je } U { ((el ,62) : 61, 62 E }.) An

example of this graph is shown in Figure 7-2. Obviously G' can be constructed in polynomial

time. Moreover, G has a proper k-coloring if and only if G' allows a broadcast schedule with

frame length I k + m. To see this, note that by the construction of graph G', (vl, v2) E E if and

only if vl, v2 E V are 2-hop neighbors in G'. Also, V' \ V forms a clique in G', and any vertex in

this clique is a 2-hop neighbor of any vertex in V since G is connected. Thus no other vertex can

transmit in the same time slot with a vertex from the clique, so any broadcast schedule in G' will

require m time slots just for vertices from the clique to transmit.

The remaining vertices in V' (i.e., vertices from V) can transmit according to any proper

coloring in G, where different time slots in the resulting broadcast schedule will correspond to

different colors in the coloring. Therefore, there is a one-to-one correspondence between proper

colorings in G and feasible broadcast schedules in G'. We see that in fact k-coloring reduces in

polynomial time to K-MSP, where K = k + m. Thus, the proof is complete.

7.4 Heuristics

In this section, we introduce and discuss several heuristics which have been applied to the

MSP-TDMA with varying degrees of success. The specific algorithm which are compared include:

Greedy Randomized Adaptive Search Procedure (GRASP) [39], GRASP with Path Relinking

[28], Reactive GRASP with Path Relinking [27], Sequential Vertex Coloring (SVC) [175], Mean

Field Annealing (MFA) [169], a Mixed Neural-Genetic Algorithm (HNN-GA) [164], and we

present a new combinatorial algorithm by Commander and Pardalos [43].

7.4.1 Combinatorial Algorithm for TDMA Message Scheduling

The inherent intractability of the problem motivates the need for efficient heuristics to

quickly provide good solutions for non-trivial instances. In this section, we describe a new


is a two-phase iterative procedure for which pseudo-code is provided in Figure 7-3. First, we

procedure ComAlgBSP(G')
1 M*<- |V |
2 X* <-
3for i = 1 to Maxlter do
4 M <- S10tMinimization(G', a~, S10tIter)
5 if M~ < M* then
6 M~* <--
7 X <- BurstMaximization(G', M~*, V*)
8 end
9if X > X* then
to X* <-X
11 end
12 end
13 return (M~*, X*, V*)
end procedure ComAlgBSP

Figure 7-3: Pseudo-code of the proposed heuristic for MSP-TDMA.

concentrate primarily on the frame length minimization portion of the MSP-TDMA by using a

greedy heuristic for graph coloring which computes near optimal solutions for the FLMP. Since

this solution will only have each station transmitting exactly once, a local improvement method

is then applied which attempts to maximize the throughput within the derived frame length. To

increase the efficiency of the procedure, the BurstMaximizat ion procedure is only entered if

the current frame length M~ is as least as small as the current best value M~*. After some specified

number of iterations, the algorithm terminates returning the best overall solution, which consists

of the frame length M~*, the total number of bursts X*, and the schedule of slot assignments V*.

Frame Length Minimization. For the first phase of the algorithm, we apply a greedy

construction heuristic to determine the value for M~, the number of time slots required for all

stations to transmit. As a result of Theorem 25, the method is based on the construction phase

of the Greedy Randomized Adaptive Search Procedure (GRASP) [157] for coloring sparse

graphs proposed by Laguna and Marti in [126]. This particular method was chosen because it

is able to quickly provide excellent solutions for the frame length. That being said, any other

coloring heuristic would work fine for the frame length minimization phase. In fact, in [175] a

method based on Sequential Vertex Coloring was used to determine the value of M~. However,

procedure S10tMinimization(G', a~, S10tIter)

2 V' <--V
3 while V' / 0) do

5 Ecount <- 00
6 for j = 1 to S10tIter do
7 Vi <- V', U <- 0, S <- 0
8 while V / 0) do
9 if U = 0) then
to RCL <- {(1 ~) 101 1' stations of max degree in V}
11 else
L2 RCL <- {(1 ~) 101 1' stations of max degree in V n U}
13 end if
14 s <- randSelect(RCL)
15 S <-- SU
16 N(s)t <- w|(S, W)E E'}
17 V <- V/(({s} U Ncs)
18 U <-U U N(s)
19 end while
20 E* <- (u, v)E E'|u, vE V'/S}
21 if |E*| < Ecount then

23 Ecount <- |E*|
24 end if
25j end for
26 V' <--V'/1
27 end while
28 return (M, V* = {VI, V2..., VM ~))
end procedure S10tMinimization

Figure 7-4: Greedy randomized heuristic for frame length minimization.

the randomized approach of the chosen method allows us to explore the search space more

thoroughly and provides several feasible solutions to work with in the throughput maximization

phase. This is because different optimal colorings will yield different solutions in the second

phase. Furthermore since sparse graphs usually contain an exponential number of optimal

colorings [1 19], the chosen method leads to a variety of solutions to explore in phase two.

Pseudo-code for this routine is given in Figure 7-4.

Our implementation of the frame length minimization heuristic is exactly as described

in [126]. The procedure takes the augmented graph G', a proportional parameter a~, and a

value S10tIter as input and creates an initial broadcast schedule one slot at a time. The value

a~E [0, 1] determines the amount of randomness, or conversely greediness that the procedure

uses. S10tIter is the number of candidate schedules for a particular slot from which the best is


Initially, the frame length M~ is initialized to 0 and V', the set of unscheduled stations

is initialized to V. The initial schedule is created in the while loop from lines 3-27. After

incrementing the frame length, the for loop from lines 6-25 is entered. In this loop, S10tIter

candidate schedules are created for the current slot VM~. Initially, V, the set of admissible

unscheduled stations is initialized to V' and U, the set of inadmissible scheduled vertices is

initialized to the empty set. S, the set of stations scheduled in the current slot, is also set to 0).

From lines 9-11 a so-called Restricted Candidate List (RCL) is constructed and contains the

(1 a~)10H' .; admissible stations of maximum degree. It is now clear how the particular value

of a~ controls the amount of randomness that is used by the algorithm. A value of a~ = 0 would

result in a simple random search, while a~ = 1 would yield a pure greedy search [152]. After the

construction of the RCL, an element s E RCL is chosen at random and scheduled in the current

slot in line 15. The sets V and U are updated and the loop continues. After the slot capacity is

maximal, the set E* is computed which contains the set of edges remaining in the graph induced

by the yet unscheduled stations. If |E*|I is less than the current minimum value Ecount, then the

current candidate slot schedule is saved in VM~ in line 22. In line 26 after S10tIter samples, the

best slot schedule is removed from the graph and the main loop repeats [126]. Finally, the frame

length M~ and the final slot schedule V* = { V, V2, .., VM} are returned to the main procedure.

The result of this procedure is a feasible solution for MSP-TDMA in which each station

is scheduled to broadcast in exactly one slot during the frame. This follows directly from the

result proven in Theorem 25. For a discussion of the computational complexity of the proposed

procedure, the reader is referred to [126].

procedure BurstMaximization(G', M~*, V*)
S X<- |V |
2 for i = 1 to M~ do
3 T <-V
4 while T / 0) do
5 T <- {vl~v 5( 1 and V se E (v, s) 5( E'}
6 8 <- randSelect(T)
7 M <- 1U { s }
8 X <-X +1
9end while
to end for
II retunm (X, V* {I 2,.,V
end procedure BurstMaximization

Figure 7-5: Throughput maximization pseudo-code.

Throughput Maximization Phase. The second phase of the proposed method attempts

to maximize the throughput beginning with the feasible solution found in the frame length

minimization phase. Clearly, the solution from the first phase will not provide an optimal

throughput in general, because each station will only be scheduled to transmit once in the frame.

Therefore, we use a randomized local improvement method to schedule each station as many

times as possible in the frame.

Pseudo-code for the throughput maximization heuristic is provided in Figure 7-5, and the

method proceeds as follows. Since each station is only scheduled once, the total number of

bursts, X is set to |V|. The main loop from lines 2-10 locally optimizes each slot in the frame.

First, the set of stations which can transmit along with those stations already scheduled in

the current slot, namely T is initialized to V. T is then updated and contains those stations v

which are not already scheduled in the current slot and are not adj acent to any station a which

is scheduled in the current slot. An element of T is then selected randomly and added to the

current slot. In line 8 the total number of bursts is incremented, and the loop repeats. The method

proceeds to the next slot when there are no stations which can transmit with those currently

scheduled, i.e., when T = 0. The method returns the total number of bursts, X and the updated

broadcast schedule V* in line 11.

7.4.2 GRASP

Recall that as described above, the Greedy Randomized Adaptive Search Procedure

(GRASP) is a two-phase iterative metaheuristic for combinatorial optimization [69, 72, 157]. In

the first phase, referred to as the construction phase, a greedy randomized initial feasible solution

is created. Then in the second phase, the initial solution is improved by the application of a local

search procedure. The solution which is best out of all GRASP iterations is returned. GRASP

has been applied to many combinatorial problems such as quadratic assignment [128, 140], job

shop scheduling [18, 7], private virtual circuit routing [156], and satisfiability [154]. GRASP

was successfully applied to the MSP-TDMA by Commander et al. in [39]. We describe the

implementation below.

Construction Phase. The construction phase for the GRASP constructs a solution

iteratively from a partial broadcast schedule which is initially empty. The stations are first

sorted in descending order of the number of one-hop and two-hop neighbors. Next, a so-called

Restricted Can2didate List (RCL) is created and consists of those greedily selected stations which

may broadcast simultaneously with the stations previously assigned to the current slot. From this

RCL a station is randomly chosen and assigned in the current slot. A new RCL is created and

another station is randomly selected. This process continues until there are no stations to put in

the RCL, at which time the slot number is incremented and the procedure is repeated recursively

for the subgraph induced by the set of all vertices whose corresponding stations have not yet been

assigned to a time slot.

Local Search. The local search phase used is a swap-based procedure which is adapted

from a similar method for graph coloring implemented by Laguna and Marti in [126]. First, the

two slots with the fewest number of scheduled transmissions are combined and the total number

of slots is now given as k = us 1, where mr is the frame length of the schedule computed in the

construction phase. Denote the new broadcast schedule as {s,7t*,;, m7' = 1, .., k, a = 1, .., N}).
Now, let the function f (s) := CE E(?rs(), wheret E(?rs() is the set of collisions in slot ?rt( f (s)

is then minimized by the application of a local search procedure as follows.

A colliding station in the combined slot is chosen randomly and every attempt is made to

swap this station with another from the remaining k 1 slots. After a swap is made, f (s) is

re-evaluated. If the result is better, that is if f (s) has a lower value than before the swap, the swap

is kept and the process repeated with the remaining colliding stations.

If after every attempt to swap a colliding station the result is unimproved, a new colliding

station is chosen and the swap routine is attempted. This continues until either a successful

swap is made or for some specified number of iterations. If a solution is improved such that

f (s) = 0, then the frame length has been successfully decreased by one slot. The value of

k is then decremented and the process is repeated beginning with the combination of the two

"smallest" slots. If the procedure ends with f (s) > 0, then no improved solution was found.

7.4.3 Sequential Vertex Coloring

In [175], Yeo et al. take a multi-obj ective optimization approach to solving the MSP-TDMA.

They implement a two-phase heuristic based on the idea of sequential vertex coloring (SVC). In

the first phase, they only consider the problem of minimizing the frame length. Then in phase 2,

the frame length is fixed with the solution from phase 1 and the utilization within the frame is


Frame Length Minimization. For this phase, the frame length minimization in the

MSP-TDMA is attacked by solving the graph coloring problem in the augmented graph. More

specifically, an algorithm based on the sequential vertex ordering method is used to solve this

problem. This is done by first ordering the stations in descending order of the number of one-hop

and two-hop neighbors. The first vertex is colored and the list of the other NV 1 vertices

are scanned downward. The remaining vertices are colored with the smallest color which has

not already been assigned to a one-hop neighboring station. The process is continued until all

vertices have been assigned a color.

Utilization Maximization. Beginning with this initial schedule, phase 2 attempts to

maximize the throughput in the TDMA frame. To maximize the utilization within the frame

whose length was determined in phase 1, an ordering method of the sequential vertex coloring

algorithm is applied. The stations are now ordered in ascending order of the the number of

one-hop and two-hop neighbors. The first ordered station is then assigned to any slots in which it

can simultaneously broadcast with the previously assigned stations. This process is repeated with

every station in the ordered list.

7.4.4 Mean Field Annealing

In 1997, Wang and Ansari [169] proposed a heuristic for the MSP-TDMA based on Mean

Field Annealing (MFA). In statistical mechanics, the physical process of annealing is used to

relax a system to the state of minimal energy. This is done by heating the solid until it melts and

then cooling it slowly so that at each temperature the particles randomly arrange themselves until

reaching thermal equilibrium.

In [116], Kirkpatrick et al. introduced a method for combinatorial problems known as

simulated annealing (Section 2.7.2). Based on the theory of the physical process, simulated

annealing was shown to asymptotically converge to the global minimum after performing a

number of so-called transitions at decreasing temperatures.

Though simulated annealing is guaranteed to converge to the global optimal solution, this

process is quite often computationally expensive. Mean field annealing, a heuristic which mimics

the idea of mean field approximation from statistical physics [150] can be employed instead.

In MFA, the stochastic process in simulated annealing is replaced by a set of deterministic

equations. Though MFA does not guarantee convergence to a global optimal solution, it

can provide an excellent approximation to an optimal solution and is much less expensive


7.4.5 Mixed Neural-Genetic Algorithm

As in the algorithm presented by Yeo et al. in [175], Salcedo-Sanz et al. [164] introduced

a two-phase heuristic based on combining both Hopfield neural networks [104] and genetic

algorithms as in [171]. As with the vertex coloring algorithm, phase one of the mixed

neural-genetic algorithm minimizes framelength and phase two attempts to maximize the

utilization within the slot.

Frame Length Minimization. The frame length minimization problem presented in [164]

is the same as described above. For the solution, a discrete-time binary Hopfield neural network

(HNN) is used. As described in [164], the HNN can be represented as a graph whose vertices are

the neurons (stations) and whose edges are the direct collisions. The graph is then mapped to the

schedule matrix S as defined above. The neurons are updated one at a time after a randomized

initialization until the system converges.

Utilization Maximization. In this phase, a genetic algorithm is used to maximize the

channel utilization within the frame length that was determined in phase one. A HNN is also

used to ensure that all constraints are satisfied. Genetic algorithms receive their name from an

explanation of the way they behave. Not surprisingly, they are based on Darwin's Theory of

Natural Selection. Genetic algorithms store a set of solutions and then work to replace these

solutions with better ones based on certain fitness criterion represented by the obj ective function


7.5 Computational Results

The proposed heuristic was coded in the C++ programming language and compiled using

MicrosoftB Visual C++ 6.0. The test machine was a PC equipped with a 1700MHz IntelB

PentiumB M processor and 1GB of RAM operating under the MicrosoftB Windows@ XP

environment. The heuristic was tested on three classical instances as well as a set of 60 random

unit disk graphs [35] with varying densities, 20 graphs each having 50, 75, and 100 nodes. The

graphs are those which were used by Butenko et al. in a prior MSP-TDMA study [39, 40].

We compared our results to those found by several heuristics from the literature, all of which

were tested on the same PC described above. As mentioned by Pitsoulis and Resende [152],

the particular value of a~ used in randomized greedy heuristics is typically determined either

empirically or chosen randomly during each iteration. Alternatively, in a Reactive GRASP the

value of a~ is tuned automatically to favor specific values that tend to produce better solutions.

Nevertheless, during our testing, we found that a value of a~ = 0.1 generally produced the

best overall solutions for the instances tested. The other parameter, S10tIter, was set to 5.

In addition, we have implemented the integer programming (IP) model for the THROUGHPUT

MAXIMIZATION PROBLEM using the Xpress-MPTAI Optimization suite from Dash Optimization

[108]. Xpress-MP contains an implementation of the simplex method [98], and uses a branch

and bound algorithm [173] together with advanced cutting-plane techniques [107, 139]. Thus not

only are we able to compare our heuristic to those in the literature, but we can also see how the

heuristics compare with the optimal solutions.

Though finding the optimal frame length is HiP-hard, we can use the IP model for the TMP

to confirm whether a frame length is optimal or not. Consider an instance of MSP-TDMA and let

Af* be the optimal frame length. Then if we set At = Af* 1 in the integer programming model

for the TMP, the resulting IP will be not yield any feasible integer solutions. In fact, the linear

programming relaxation could also be infeasible; thus implying the particular instance of the TMP

is also infeasible. The proposed heuristic was first tested using three examples first introduced by

Wang and Ansari in [169] which have since become the de facto test cases for TDMA broadcast

scheduling algorithms. These examples include networks of varying densities with 15, 30, and 40

stations. The graphs of the networks can be seen in Figure 7-6.

Table 7-1 provides the optimal solutions for the three aforementioned networks as well as

the heuristic solutions found by our combinatorial algorithm (ComAlg), the GRASP from [39],

the Mixed Neural-Genetic Algorithm (HNN-GA) proposed in [164], the Mean Field Annealing

(MFA) method from Wang and Ansari [169], and the Sequential Vertex Coloring (SVC) heuristic

from [175]. The solutions are reported as (X, Af). Notice that the proposed algorithm found the

optimal solution for each of the three instances. The average computation time required for these

instances by our method was 1.375s. The average time required by Xpress-MP to compute the

optimal solutions was 3411.4 seconds, with the 30 station network taking 10212 seconds. Next, in

order to test the scalability of the new method and evaluate its performance for general networks,

we tested the algorithms on the 60 random graphs from [39].

4 -~ 9 10 -

8 13 -

Figure 7-6: Benchmarki TDMA test, cases. (a) 15 station networks. (b) 30 station networks. (c) 40 station

Table 7-1: Comparison of solutions for the benchmark instances from Wang and Ansari.
Stations Optimal Soln ComAlg GRASP HNN-GA MFA SVC
15 (20,8) (20,8) (20,8) (20,8) (18,8) (18,8)
30 (36,10) (36,10) (36,10) (35,10) (39,12) (37,11)
40 (69,8) (69,8) (65,8) (67,8) (71,9) (60,8)

The comparative results of the proposed algorithm against the best solutions computed by

Xpress-MP after 3600 seconds, as well as the aforementioned heuristicsl on the 50 station

graphs from [39] are given in Table 7-2. The first column represents the instance name followed

by the density of the graph G'. Notice that the solutions from the new method (ComAlg) are

at least as good as any other heuristic for all of these instances. Specifically, the new method

provides better solutions for 15 of the 20 instances. The asterisk implies that the reported solution

is optimal. For these instances, the new algorithm found optimal solutions for 401' of the test

cases. The average frame utilization is also reported at the bottom of the table. The utilization p,

provides a measure of the efficiency of a broadcast schedule and is computed as follows

p := .(7-7)

We see that for the 50 station networks, the proposed algorithm has an average channel utilization

that is 101 'II .' greater than the other heuristics. The average optimality gap for the throughput

maximization phase was 1.921 The average computation time for our algorithm on these

instances was 2.8 seconds.

The comparative solutions for the 75 station networks are given in Table 7-3. Notice that our

method outperforms the other heuristics in the literature on every instance. For these networks,

the proposed algorithm has an average channel utilization that is 8.1 .< greater than the other

methods. The heuristic required on average 6.62 seconds to find the target solution, and as with

the 50 station networks, optimal frame lengths are achieved for all instances. Furthermore, the

1 The MFA algorithm of [169] was not available to the authors for testing.

Table 7-2: Comparison of optimal and heuristic solutions for graphs with | V| = 50 stations. An *
indicates that the solution is optimal, while a t indicates the solution is the best found by Xpress-MP after
3600s. Solutions are reported as (X, M~).
Instance Density Xpress-MP ComAlg GRASP HNN-GA SVC
50r20i6 0.1136 (146,10) (145,10) (143,10) (145,10) (111,10)
50r20i2 0.0824 (86,6) (86,6)* (84,6) (86,6)* (82,6)
50r20i3 0.1040 (85,6) (85,6)* (83,6) (85,6)* (60,7)
50r20i7 0.0872 (90,6) (89,6) (87,6) (89,6) (52,6)
50r20i5 0.0968 (107,7) (107,7)* (107,7)* (105,7) (64,8)
50r30il 0.1728 (78,8) (76,8) (74,8) (75,8) (54,9)
50r30i2 0.2122 (77,9) (75,9) (81,10) (70,9) (73,10)
50r30i3 0.1960 (84,9) (84,9)* (78,9) (78,9) (78,10)
50r30i4 0.2048 (74,8)t (71,8) (67,8) (67,8) (60,10)
50r30i5 0.2096 (82,9) (79,9) (76,9) (84,10) (89,11)
50r40il 0.3048 (76,12) (74,12) (73,12) (71,12) (58,14)
50r40i2 0.3680 (83,14)t (80,14) (77,14) (77,14) (83,16)
50r40i3 0.3408 (76,12)t (76,12)* (80,13) (77,13) (56,15)
50r40i4 0.3712 (81,15) (81,15)* (80,15) (76,15) (81,17)
50r40i5 0.3208 (71,12) (70,12) (67,12) (65,12) (55,14)
50r50il 0.4280 (72,17) (72,17)* (71,17) (75,18) (61,19)
50r50i2 0.4640 (61,15)t (61,15)t (65,16) (68,17) (55,17)
50r50i3 0.4480 (66.15)t (66,15)t (64,15) (65,16) (56,17)
50r50i4 0.4376 (70,15) (70,15)* (72,16) (72,16) (79,18)
50r50i5 0.4088 (55,14)t (55,14)t (58,15) (56,15) (61,18)
Avg Soln 0.2603 (80. 1,10.95) (80. 1,10.95) (79.35,11.2) (79.3,11.35) (68.4,12.6)
Avg Util -I 0. 1463 0. 1463 0. 1417 0. 1397 0. 1086

solutions from our method are always within 101' of optimal solutions, with an average gap of

Finally, the solutions for the 100 station networks are given in Table 7-4. Once more, the

new algorithm finds solutions which are superior to the other heuristics for each instance. The

utilization was an average of 10. 17' higher than the other algorithms. The average computation

time was 12.17 seconds, with reported gaps of less than 101' of the best solution found by

Xpress-MP after 3600 seconds. Notice also that Xpress-MP was unable to compute a solution

superior to the proposed heuristic for 100r50i3 and 100r50i6. Each of these instances were ran

for 10000 seconds and Xpress-MP was unable to compute a feasible solution in the frame length

achieved by our method.

Table 7-3: Com arison of optime I solver and heuristic solutions for the 75 station networks.
Instance Density Xpress-MP ComAlg GRASP HNN-GA SVC
75r20il 0.0988 (145,8) (139,8) (135,8) (136,8) (161,10)
75r20i2 0.1038 (122,8)t (119,8) (113,8) (112,8) (79,10)
75r20i3 0.1159 (113,7)t (108,7) (139,9) (121,8) (150,10)
75r20i4 0.0946 (116,7) (114,7) (109,7) (111,7) (84,8)
75r20i5 0.0988 (145,8) (138,8) (131,8) (135,8) (161,10)
75r30il 0.1927 (114,12) (110,12) (117,13) (117,13) (91,13)
75r30i2 0.1867 (110,11) (105,11) (109,12) (101,11) (94,12)
75r30i3 0.2190 (140,15) (133,15) (132,15) (132,15) (81,17)
75r30i4 0.2009 (142,13)t (133,13) (127,13) (128,13) (144,15)
75r30i5 0.1927 (119,12)t (111,12) (106,12) (108,12) (89,12)
75r40il 0.3328 (105,17)1 (103,17) (104,18) (113,19) (79,20)
75r40i2 0.2980 (108,16)t (106,16) (109,17) (115,18) (86,19)
75r40i3 0.3403 (112,19)t (109,19) (105,19) (103,19) (87,20)
75r40i4 0.3492 (126,20)t (118,20) (124,21) (119,21) (79,24)
75r40i5 0.3143 (104,16)t (97,16) (100,17) (109,18) (114,20)
75r50il 0.4587 (110,23)1 (106,23) (107,24) (108,25) (123,29)
75r50i2 0.4622 (102,23)t (97,23) (99,24) (104,25) (108,27)
75r50i3 0.4807 (106,24)t (102,24) (104,25) (111,27) (114,29)
75r50i4 0.4750 (121,26)t (115,26) (112,26) (110,26) (102,28)
75r50i5 0.5088 (106,25)t (104,25) (111,27) (107,27) (106,28)
Avg Soln 0.2686 (118.45,15.5) (113.25,15.5) (114.65,16. 15) (115,16.4) (106.6,18.05)
Avg Util -0. 1019 0.0974 0.0947 0.0935 0.0787

As mentioned above, the strategy of first minimizing the frame length and then attempting to

maximize the throughput within this frame is a common approach [39, 164, 175]. In particular,

in [175], the authors propose a heuristic based on sequential vertex coloring to provide a

feasible frame length. Then, a greedy heuristic is used to maximize the throughput. Similarly,

Salcedo-Sanz et al. propose a hybrid heuristic which minimizes the frame length using a neural

network and then maximizes the throughput using a genetic algorithm [164].

The algorithm proposed in [39] based on the GRASP metaheuristic described above, uses a

slightly different strategy to provide approximate solutions for the MSP-TDMA. In this algorithm,

the construction phase creates a solution iteratively in a similar manner to the one proposed

in this chapter. The maj or difference is that the constructor in [39] does not contain the slot

candidate construction loop (Figure 7-4, lines 6-25). Instead the final station assignments in

each slot are taken as the first produced during the greedy randomized construction. This is

equivalent to setting S10tIter equal to 1 in the S10tMinimization method above. By setting

S10tIter > 0, the proposed method is more likely to produce better solutions during the frame

length minimization phase.

The local search used in the GRASP in [39] not only attempts to maximize the throughput

within the frame created during the construction phase, but also tries to reduce the frame length

further. This method was adapted from the GRASP for coloring sparse graphs of Laguna and

Marti in [126]. Here the two slots with the fewest broadcasts are combined creating a new

(infeasible) schedule with one less slot than construction solution. The set of stations which cause

message collisions as a result of the slot combination is determined. For each station causing

a collision, every attempt is made to swap them with another station from the remaining slots.

If the swap reduces the number of collisions it is kept and the remaining colliding stations are

considered. If all collisions are successfully averted, the process repeats with the combination

of two more slots. This contrasts with the proposed method in that after the frame length is

determined by the more picky S10tMinimization method, it is fixed for the current iteration.

Experimental analysis shows that our algorithm is superior to the other heuristics in the

literature. For all 63 instances tested, the method found solutions at least as good as any of the

other algorithms from the literature for all of the networks, outperforming them on 56 cases.

Also, we see that attempting to solve large-scale instances optimally is impractical. However,

our heuristic required only 7.49 seconds on average to find solutions that are within 4.15' of the

average best solution found by the commercial IP solver in 3600 seconds.

7.6 Concluding Remarks

In this chapter, we described and implemented several heuristics for the MESSAGE

SCHEDULING PROBLEM ON TDMA NETWORKS. In addition, we have implemented an optimal

solver using Xpress-MP [108]. The MSP-TDMA is an important problem that occurs in wireless

mesh networks regarding efficiently scheduling collision free broadcasts for the network stations.

The obj ective of the MSP-TDMA is two-fold. First, the number of slots required to schedule all

stations is minimized. Then the throughput is to be maximized by scheduling as many stations as

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 B
2~B B
3 B B
4 B B B
5 B B B
6 B B B
8 B B B

Main1 2 3 4 5 6 ? 8 9 10I 11 12 13 14 IS 16 17 13 li 23 11 22 21 i 5 26 2T 28 29 2
1 BB B
2 B B B
3 BB B
4 B 8 B
6 B B B B
6 B B B
? BB B B
8 B B B
9 B 6 B B
10 B B B 8 B


1121 145 1 781 1819 11 2III131 151 1 718IBIiiI I19 1 21 lp2]lar 5 2 7 2 5 9 112 5 5 5 5 3i5 5@ls I~

I e a B B Bs B
3 8 B B B 8

5 8 1 b a 8 B a
I B I I a B B B

I B 8I B 8 B 8


Figure 7-7: Example GRASP broadcast schedules for the networks given in Figure 7-6: (a) 15
station network, (b) 30 station network, (c) 40 station network.


Throughout this dissertation, we focused on optimization problems in telecommunication

systems, with a particular emphasis on wireless ad hoc networks operating in a military

environment. We examined two broad classes of problems; namely, that of ensuring

communication on the network, and conversely at problems of denying service on the network.

The problems presented all have similar traits. For example, they are all modeled as discrete

optimization problems on graphs. Furthermore, as we saw all the problems we examined were

HiP-hard. We presented an in-depth look at the computational complexity of each problem and

examined ways of designing efficient algorithms for each.

Chapter 1 provided an introduction to the dissertation with a brief description of the

maj or contributions and a description of the problems to follow in the subsequent chapters. In

Chapter 2, we presented an introduction to global optimization which included many theorems

and basic definitions which were applied in the later chapters. This chapter provided a foundation

for the work to follow.

The subj ects of the next three chapters studied methods of denying service on

telecommunication networks. These problems help to identify weaknesses and vulnerabilities in

networks. In Chapter 3, we began with the study of jamming wired telecommunication networks.

We presented two formulations and analyzed their computational complexity. Next we provided

heuristics which provided excellent solutions for real and randomly generated data sets in a

fraction of the time required by a commercial software package.

In Chapter 4, the general WIRELESS NETWORK JAMMING PROBLEM was introduced. This

problem is an extension of the CRITICAL NODE PROBLEM studied in Chapter 3. We examined

several variations of the problem and provided integer programming formulations for each.

Furthermore, we provided formulations which included percentile constraints. The case studies

presented showed that the addition of the percentile risk measures provided excellent solutions

with a significant reduction in cost. Heuristic algorithms were also presented and a computational

study was presented. For each problem presented in this chapter, we assumed that there as a

priori knowledge of the network to be j ammed. We saw that even with this seemingly generous

assumption the problems remained HiP-hard.

The work in Chapter 5 relaxed this assumption and considered the problem of jamming

a network when no information was assumed other than the general area known to contain the

network. We considered a subproblem of placing the j amming devices on a lattice overlaying the

region. A rigorous analysis followed in which we derived upper and lower bounds on the optimal

number of jamming devices required to suppress the network. We showed that by considering

the cumulative effect of the j amming devices, that our result was superior to the classical method

of covering a region in the plane with uniform circles. Furthermore, a convergence result was

provided showing that the bounds are tight within a constant. To conclude this chapter we

presented a randomized local search algorithm which began with the lower bound value derived

and attempted to minimize the number of devices need to cover the region. Experimental results

indicated that the heuristic was able to reduce the number of jamming devices by approximately



(CCPMANET) was the topic of Chapter 6. This problem is concerned with determining the

routes for a set of mobile agents in such a manner that communication amongst the agents is

maximized. We examined several obj ective functions and compared their relative advantages

and provided an integer programming formulation. We then provided a computational study

and proved that the problem is HiP-hard via a reduction from the 3-SATISFIABILITY problem.

Further, we proved that it is HiP-hard to compute an optimal solution at each discrete time step.

Next, we derived several heuristics and provided an extensive computational analysis.

In Chapter 7, we took a closer look at the particular communication devices used by

the agents in the CCPMANET problem. In particular, we examined the TDMA MESSAGE

SCHEDULING PROBLEM. TDMA is a type of time-division multiplexing where multiple users

share the same frequency channel by dividing the signal into different time slots. The users are

then scheduled to broadcast in a set of time slots such that there i s no interference by users which

broadcast in the same slot. We began by examining the recognition version of the problem and

showed that it is HP~-complete. WVe followed this by designing several heuristics and comparing

their effectiveness against other heuri sti cs from the literature.

As telecommunication systems evolve ever so rapidly, there are as many direction for future

research as one can imagine. At the conclusion of each chapter we indicated several problems

and extensions which could fll a11 .:- from the specific work at hand. Of course, the quest for

efficient algorithms with better worst case complexity will always lie at the forefront for all

the problems considered. Also, the development of tight upper and lower bounds will certainly

aid all future endeavors for each problem. It is my hope that my work represents the current

state-of-the-art for the problems presented, and that my efforts will help our military perform

better as they face the daunting task of defending our freedoms wherever they are called.


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Clayton W. Commander was born in Ft. Walton Beach, Florida, on August 2.3, 1~1-` He

was raised in nearby Niceville, Florida, and graduated from Niceville High School in t'r r Ir

After receiving an Associate of Arts degree from Okaloosa-Walton Community College, Clayton

enrolled in the Department of Mathematics at the University of Florida in August of 2001 In

May :'(1( if he graduated summa cum laude and began working for the United States Air Force.

In January _'s~ 1, while working at Eglin Air Force Base, Clayton entered graduate school

in the Department of Industrial and Systems Engineering at the University of Florida and began

studying optimization with Professor Panos Pardalos. The happiest day of Clayton's life came on

June 18, '1 II I when he married the love of his life, Leah Susi. Hie received a master's degree in

December 2005 and earned his Ph.D. in August 2007. Clayton and Leah live happily with their

two chihuahuas Reina and Isabelle in Niceville, FL.




c r 2007ClaytonWarrenCommander 2


Tomyfamily. 3


ACKNOWLEDGMENTS First,ImustthankmyadvisorPanosPardalos.Hehasbeenani ncrediblementor,leader,and friendtomesincethedaywemet.Hisexcitementandpassionf orlife,research,andfamilyhave hadaprofoundeffectonmeandhaveencouragedmegreatly.He willalwayshaveaspecialplace inmyheart. Myappreciationgoestomycommitteemembers,StanUryasev, J.ColeSmith,andWilliam Hagerfortheirtimeandhelpfulideasthatguidedmealongth eway.Iwouldalsoliketothank themembersofthegraduatecommitteeFaridAitSahlia,Elif Akcali,andEdwinRomeijnfor givingastudentfromwhomtheyhadnothingtogainanotherch ance. NextIwishtothankmywonderfulco-authorsandcollaborato rsforworkingwithme andforhelpingmelearnhowtobearesearcher:AshwinArulse lvan,SergiyButenko,Lily Elefteriadou,PaolaFesta,MichaelJ.Hirsch,CarlosA.S.O liveira,MichelleRagle,Mauricio G.C.Resende,ValeriyRyabchenko,OlegShylo,MarcoTsitse lis,StanUryasev,YinyuYe, andGrigoryZharshevsky.IwouldliketothankJonathanKing forinstillinginmealoveof mathematicsandanappreciationforclearlywrittenmathem aticaldiscourse.Iamparticularly gratefultoMauricioG.C.ResendeofAT&TLabsResearchforh iswonderfulcollaborationsand helpovertheyears.Heiseverybitasgraciousasheisbrilli ant.Finally,IamgratefultoClaudio MenesesandOnurSereffortheirthoughtfuladvice. IamtrulygratefultotheUnitedStatesAirForceforsupport ingandnancingmy educationalendeavors.ParticularthanksgotoRobMurphey ,DavidJeffcoat,MichelleWhite, andmanyothersattheAirForceResearchLaboratoryforthei rsupport.ThanksgotoDon Grundelwhoalwaysgavegoodadviceandkeptmeonthestraigh tandnarrow. Finally,butcertainlynotleast,mymostheartfeltappreci ationgoestomyfamily.Ithankmy parentsandmygrandmotherwhoalwayslistenedandencourag edme.Ithankmyparents-in-law foralwaysgivingmeaplacetostay.Finally,Ithankmybeaut ifulwifeLeahwhohasbeenmy constantsourceoflove,passion,andinspiration. !ylydwdwydwdlyn' 4


TABLEOFCONTENTS page ACKNOWLEDGMENTS ....................................4 LISTOFTABLES .......................................8 LISTOFFIGURES .......................................10 ABSTRACT ...........................................13 CHAPTER 1INTRODUCTION ....................................15 2GLOBALOPTIMIZATIONISSUES ...........................16 2.1Introduction .....................................16 2.2Idiosyncrasies ....................................16 2.3FundamentalResults ................................17 2.4DiscreteOptimization ................................21 2.5ComputationalComplexity .............................22 2.6UpperandLowerBounds ..............................24 2.7AlgorithmsforOptimizationProblems .......................28 2.7.1ExactMethods ................................28 2.7.2Heuristics ..................................31 2.8ConcludingRemarks ................................36 3JAMMINGCOMMUNICATION NETWORKSVIACRITICALNODEDETECTION ..................38 3.1Introduction .....................................38 3.2ProblemFormulations ................................40 3.2.1CriticalNodeProblem ............................40 3.2.2CardinalityConstrainedProblem ......................44 3.3HeuristicsforCriticalNodeProblems .......................46 3.3.1CNPHeuristic ................................46 3.3.2CC-CNPHeuristic ..............................49 3.3.3GeneticAlgorithmfortheCC-CNP .....................50 3.4ComputationalResults ...............................53 3.4.1CNPResults .................................53 3.4.2CC-CNPResults ...............................55 3.5ConcludingRemarks ................................59 4THEWIRELESSNETWORKJAMMINGPROBLEM .................62 4.1Introduction .....................................62 4.2DenitionsandAssumptions ............................63 5


4.3DeterministicFormulations .............................64 4.3.1CoverageApproach .............................64 4.3.2ConnectivityFormulation ..........................65 4.4DeterministicSetupwithPercentileConstraints ..................72 4.4.1Value-at-Risk(VaR)andConditionalValue-at-Risk( CVaR) ........72 4.4.2PercentileConstraintsandtheWNJP ....................76 4.5CaseStudiesandAlgorithms ............................80 4.5.1CoverageFormulation ............................80 4.5.2ConnectivityFormulation ..........................82 4.6ConcludingRemarks ................................84 5JAMMINGCOMMUNICATION NETWORKSUNDERCOMPLETEUNCERTAINTY .................85 5.1Introduction .....................................85 5.2Descriptions,Assumptions,andDenitions ....................86 5.3ProblemFormulation ................................87 5.4UpperandLowerBounds ..............................90 5.5HeuristicforUncertainJamming ..........................102 5.6ConcludingRemarks ................................105 6COOPERATIVECOMMUNICATION INMOBILEADHOCNETWORKS ...........................106 6.1Introduction .....................................106 6.2DiscreteFormulations(CCPMANET-D) ......................109 6.3AlgorithmsforCCPMANET-D ...........................113 6.3.1ConstructionHeuristic ...........................113 6.3.2LocalImprovement .............................115 6.3.3One-PassHeuristic .............................117 6.3.4GreedyRandomizedAdaptiveSearch ...................119 6.3.5ComplexityoftheHeuristic .........................123 6.3.6ComputationalExperiments .........................125 6.4AContinuousFormulation(CCPMANET-C) ...................131 6.5ConcludingRemarks ................................136 7THETDMAMESSAGESCHEDULINGPROBLEM ..................139 7.1Introduction .....................................139 7.2ProblemDescription ................................140 7.3ComputationalComplexity .............................142 7.4Heuristics ......................................145 7.4.1CombinatorialAlgorithmforTDMAMessageScheduling .........145 7.4.2GRASP ...................................150 7.4.3SequentialVertexColoring .........................151 7.4.4MeanFieldAnnealing ............................152 7.4.5MixedNeural-GeneticAlgorithm ......................152 6


7.5ComputationalResults ...............................153 7.6ConcludingRemarks ................................159 8CONCLUSION ......................................162 REFERENCES .........................................165 BIOGRAPHICALSKETCH ..................................177 7


LISTOFTABLES Table page 2-1Growthratesofseveralpolynomialandnon-polynomialf unctions. ...........24 3-1ResultsofIPmodelandheuristiconterroristnetworkda ta. ...............53 3-2ResultsofIPmodelandheuristiconrandomlygenerateds calefreegraphs. .......56 3-3ResultsofIPmodelandheuristicsonterroristnetworkd ata. ..............57 3-4ResultsoftheIPmodelandgeneticalgorithmandthecomb inatorialheuristicon randomlygeneratedscalefreegraphs. ..........................58 3-5Comparativeresultsofthegeneticalgorithmandthecom binatorialheuristicwhen testedonthelargerrandomgraphs.Duetothecomplexity,we wereunableto computethecorrespondingoptimalsolutions. ......................59 4-1Optimalsolutionsusingthecoverageformulationwithr egularandVaRconstraints. ..80 4-2Optimalsolutionsusingthecoverageformulationwithr egularandVaR,andCVaR constraints. ........................................81 5-1Comparing N 2 N 1 forvariousvaluesof k ..........................95 5-2Numericalresultsareprovidedforseveralregionswith variousrequiredjamming levels.TheUpperBound,LowerBound,OptimalGrid,andLoca lSearchcolumns providethenumberofjammingdevicesrequiredforthecorre spondingregion accordingtothetheoremspresentedandtheproposedlocals earch.ThePercent Decreaseshowsthesavingswhencomparingthelocalsearcht otheoptimalgrid approach. .........................................105 6-1Comparativeresultsbetweenshortestpathsolutionsan dheuristicsolutions. ......118 6-2Threeinstanceswithdifferentsetsofagentson50nodeg raphsaregiven.Thevalue inthe UBound columnwasfoundusingCorollary1. ...................127 6-3Threeinstanceswithdifferentsetsofagentson75nodeg raphsaregiven.Thevalue inthe UBound columnwasfoundusingCorollary1. ...................128 6-4A 100 nodeinstancewithsolutionswithradiusvaryingfrom 1 to 5 units.Thevalue in UBound wasfoundusingCorollary1. .........................129 6-5AveragesolutionvaluesforGRASPandGRASPwithpath-re linkingon50node graphs. ...........................................130 6-6ComparativesolutionsofGRASPandGRASPwithpath-reli nkingon75nodegraphs. 130 6-7ResultsofGRASPandGRASPwithpath-relinkingon100nod egraphs. ........131 8


7-1Comparisonofsolutionsforthebenchmarkinstancesfro mWangandAnsari. .....156 7-2Comparisonofoptimalandheuristicsolutionsforgraphswi th j V j =50 stations.Anindicatesthatthesolutionisoptimal,whileayindicatesthesolutionisthebestfoundby Xpress-MPafter 3600 s.Solutionsarereportedas ( X;M ) ..................157 7-3Comparisonofoptimalsolverandheuristicsolutionsforth e 75 stationnetworks. .......158 7-4Comparisonofoptimalsolverandheuristicsolutionsforne tworkswith j V j =100 stations. ..160 9


LISTOFFIGURES Figure page 2-1Noticethattheroundedintegersolutionisnotoptimal. .................22 2-2Visualizationofcomplexityclasses. ...........................23 2-3Pseudo-codeforagreedyalgorithmwhichmakeschangeus ingtheminimumnumber ofcoins. ..........................................32 2-4GRASPformaximization .................................33 2-5Genericsimulatedannealingmaximizationalgorithm. ..................35 2-6Pseudo-codeforgenericgeneticalgorithm. ........................36 3-1ConnectivityIndexofnodesA,B,C,Dis3.ConnectivityI ndexofE,F,Gis2. ConnectivityIndexofHis0. ...............................45 3-2Heuristicfordetectingcriticalnodes. ...........................46 3-3Localsearchalgorithmforcriticalnodeheuristic. ....................48 3-4Heuristicwithlocalsearchfordetectingcriticalnode s. .................49 3-5HeuristicfortheCARDINALITYCONSTRAINEDCRITICALNODEPROBLEM. .....50 3-6Pseudo-codeforagenericgeneticalgorithm. .......................51 3-7Exampleofthecrossoveroperation.Inthiscase, CrossProb =0 : 65 ..........52 3-8TerroristnetworkcompiledbyKrebs. ..........................54 3-9Optimalsolutionwhen k =20 ..............................55 3-10Optimalsolutionwhen L =4 ..............................57 4-1ConnectivityIndexofnodesA,B,C,Dis3.ConnectivityI ndexofE,F,Gis2. ConnectivityIndexofHis0. ...............................66 4-2GraphicalrepresentationofVarandCVaR. ........................73 4-3Casestudy1.Theplacementofjammersisshownwhenthepr oblemissolvedusing theoriginalandVaRconstraints. .............................81 4-4Casestudy1continued.Theplacementofjammersisshown whentheproblemis solvedusingVaRandCVaRconstraints. .........................82 4-5CaseStudy2:Originalgraph. ...............................83 4-6Acomparisonofthepercentileconstrainedsolutions.I nbothcases,thetriangles representtheplacementofjammingdevices. .......................83 10


5-1Uniformgridwithjammingdevices ...........................88 5-2Theleastcoveredpointisshowninthelowerleftgridcel l. ...............89 5-3SquareDecomposition ..................................89 5-4EquivalentPoints .....................................90 5-5Cumulativeemanationofjammingdevices. .......................91 5-6IntegralLowerBound. ..................................92 5-7IntegralUpperBound. ..................................97 5-8Comparisonofthelowerandupperbounds. .......................100 5-9Pseudo-codefortherandomizedlocalsearchforuncerta injamming. ..........103 5-10Exampleofheuristicversusuniformplacement. .....................104 6-1Pseudo-codefortheshortest-pathconstructionheuris tic. ................114 6-2Pseudo-codefortheHillClimbingintensicationproce dure. ..............116 6-3Pseudo-codefortheone-passheuristic. ..........................117 6-4GRASPformaximization .................................119 6-5GreedyrandomizedconstructorforCCPMANET-D. ...................121 6-6LocalsearchforCCPMANET-D. ..............................122 6-7Path-relinkingsubroutine. .................................124 6-8GRASPwithpath-relinkingformaximization. ......................126 6-9EvolutionofGRASP+PRsolutionvalueson50nodegraphsa sthecommunication radiusincreasesfrom1to5units. ............................132 6-10EvolutionofGRASP+PRsolutionvalueson75nodegraphs asthecommunication radiusincreasesfrom1to5units. ............................133 6-11EvolutionofGRASP+PRsolutionvalueson100nodegraph sasthecommunication radiusincreasesfrom1to5units. ............................134 6-12Theheavysidefunction H 1 ................................135 6-13Alternateobjectivefunction H 2 .............................136 6-14Secondalternateobjectivefunction H 3 .........................137 11


7-1CounterexampletotheclaimofWang&Ansarithatoptimal graphcoloringcanbe foundbyrecursivelyndingamaximumindependentsetandre movingitfromthe graph. ...........................................143 7-2Constructionofgraph G 0 from G ............................144 7-3Pseudo-codeoftheproposedheuristicforMSP-TDMA. .................146 7-4Greedyrandomizedheuristicforframelengthminimizat ion. ..............147 7-5Throughputmaximizationpseudo-code. .........................149 7-6BenchmarkTDMAtestcases. ..............................155 7-7ExampleGRASPbroadcastschedulesforthenetworksgive ninFigure7-6:(a)15 stationnetwork,(b)30stationnetwork,(c)40stationnetw ork. .............161 12


AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy OPTIMIZATIONPROBLEMSINTELECOMMUNICATIONS WITHMILITARYAPPLICATIONS By ClaytonWarrenCommander August2007 Chair:PanagoteM.PardalosMajor:IndustrialandSystemsEngineering Inrecentdecades,optimizationproblemsintelecommunica tionsystemshavebeenthefocus ofanintensiveamountofresearch.Theseproblemsareimpor tantforseveralreasonsincluding speedandqualityofcommunicationamongothers.Inthisdis sertation,wepresentseveral problemsarisingintelecommunicationnetworksinmilitar yapplications.Severalproblems weconsiderinvolvewirelesscommunicationnetworks.Thes enetworksareanextraordinarily convenientmethodofcommunication.However,alongwithth isconveniencecomesamyriadof complicatedproblemsthatmustbeaddressedtopreservethe attractivefeaturesofthenetworks. Furthermore,problemsarisinginadversarialenvironment sdifferfromthoseinconventional settings,inthattimeisusuallyacriticallyconstrainedf actor.Thisistroublesomebecausemany oftheproblemsaredifculttosolveandwouldrequireatrem endousamountoftimetocompute theoptimalsolution.Howeverinabattlespaceenvironment ,timespentcomputingasolution andnotghtingtheenemyleadstoapotentiallossofmaterie landlives.Thusfortheproblems studied,wewillfocusagreatdealofattentionondesigning heuristicalgorithmswhichare capableofcomputingnearoptimalsolutionsveryefcientl y. Wewillconsidertwoclassesofproblemsinvolvingtelecomm unicationnetworks.Therst classfocusesondenyingcommunicationonanetworkanddest royingitsfunctionality.The otherclasshastheobjectiveofguaranteeingcommunicatio nonanetwork.Atrstglance,these twosetsappeartobepolaroppositesofoneanother.However ,withanyemergingtechnology studieswhichassessbothvulnerabilitiesandcapabilitie smustbeperformedinordertoachieve 13


asystemwhichwillnotfailinitsintendedoperationalenvi ronment.Ourgoalistoshow howtheseproblemscanbeformulatedandsolvedusingtoolsf romglobalandcombinatorial optimization.Fortheproblemsconsidered,weexaminethec omputationalcomplexityand examineseveralmathematicalprogrammingformulations.T henwepresentseveralalgorithms andexamineextensivecomputationalresultscomparingthe ireffectiveness.Finally,weconclude bysummarizingourworkandindicatingfuturedirectionsof research. 14


CHAPTER1 INTRODUCTION Optimizationproblemsintelecommunicationsystemshaveb eenthefocusofan intensiveamountofresearchinrecentdecades[ 135 155 ].Theseproblemsareimportant forseveralreasonsincludingspeedandqualityofcommunic ationandcostrelatedissues. Inthisdissertation,wepresentseveralproblemsarisingi nmilitaryapplicationsinvolving telecommunicationnetworks.Severalproblemsweconsider involvewirelessnetworks.These networksareanextraordinarilyconvenientmethodofcommu nication;however,alongwith thisconveniencecomesamyriadofcomplicatedproblemswhi chmustbeaddressedinorder topreservetheattractivefeaturesofthenetworks.Furthe rmore,problemsarisinginadversarial environmentsdifferfromconventionalproblemsinthattim eisusuallyacriticallyconstrained factor.Thispresentssomewhatofaproblembecausemanyoft heproblemsareextremely difculttosolveandwouldrequireatremendousamountofti metocomputetheoptimal solution.Howeverinabattlespaceenvironment,timespent computingasolutionandnotghting theenemyleadstoapotentiallossofmaterielandlives.Thu sthroughoutthisdissertationwewill focusagreatdealofattentionondesigningheuristicalgor ithmswhicharecapableofcomputing nearoptimalsolutionsveryefciently. Theremainingchaptersofthisdissertationpresenttheres ultsofmyeffortstomodeland solvemanyimportanttelecommunicationproblemsfacingth emilitaryintheeverevolving globalwaronterrorism.Wewillconsidertwoclassesofprob lemsinvolvingtelecommunication networks.Therstclass(Chapters 3 4 and 5 )focusondenyingcommunicationonanetwork anddestroyingitsfunctionality.Conversely,theproblem sinChapter 6 andChapter 7 havethe objectiveofguaranteeingcommunicationonanetwork.Atr stglance,thesetwosetsappearto bepolaroppositesofoneanother.However,withanyemergin gtechnologystudieswhichassess bothvulnerabilitiesandcapabilitiesmustbeperformedto achieveasystemwhichwillnotfailin itsintendedoperationalenvironment.Ourgoalistoshowho wtheseproblemscanbeformulated andsolvedusingtoolsfromglobalandcombinatorialoptimi zation[ 74 106 127 ]. 15


CHAPTER2 GLOBALOPTIMIZATIONISSUES 2.1Introduction Overthepast60years,OperationsResearch(OR)hasemerged asoneofthemostexciting, fast-paced,andinterdisciplinaryeldsofmathematics.S inceitsrebirthduringWorldWarII, ORhasturnedintoafascinatingsubjectwhichcrossesalldi videsfromrealanalysis,probability, statistics,economics,theoreticalcomputerscience,and biologyinanattempttosolvesomeof themostcomputationallydifcultproblemsknowntoexist. Asmentionedin[ 98 ],ORwasrstformalizedduringWorldWarIIwhensupplieswe re limitedandneededtobeallocatedtothealliedforcesovers eas.ORteamswerefundamentalin developingmethodsforusingradarwhichwascrucialinthea llieswinningtheairwar.Later, researchersdevelopedmethodsforoptimallytransporting convoysandderivedmethodsfor trackingsubmarinesthusleadingtosuccessinthePacicth eater.Theoriginalnameoftheeld was MilitaryOperationsResearch ;however,duetothesuccessofthemethodsderivedduringth e war,scientistsandengineersbeganapplyingthesetechniq uestootherproblemsinmathematics andindustrialengineering.Theword military wassubsequentlydroppedbecauseofthis. Sincetheearly1950's,researchershavebeenexpandingthe techniquesandmethodsofOR. ReminiscentofthetimeofGaussandEuler,scientistsarema kingcontributionsatincredible ratesineldsrangingfromfacilitylocationproblemstoth emappingofthehumangenome. Withtheadventofthedigitalcomputer,algorithmsarenowa bletobeimplementedproviding thecapabilitytosolveproblemsneverbeforethoughttract able.Inthischapterwepresentthe foundationofglobaloptimization.Thiswillprovidethene cessarytoolsforusasweinvestigate theproblemspresentedinthesucceedingchapters. 2.2Idiosyncrasies Inthissubsection,weintroducethesymbolsandnotationsw ewillemploymostfrequently throughoutthisdissertation.Denoteagraph G =( V;E ) asapairconsistingofasetofvertices V ,andasetofedges E .Letthemap w : E 7! R beaweightfunctiondenedonthesetof 16


edges.Wewilldenoteanedge-weightedgraphasapair ( G;w ) .Thuswecaneasilygeneralize anun-weightedgraph G =( V;E ) asanedge-weightedgraph ( G;w ) ,bydeningtheweight functionas w ij := 8>><>>: 1 ; if ( i;j ) 2 E; 0 ; if ( i;j ) 62 E: (2–1) Weusethesymbol“ b := a ”tomean“theexpression a denesthe(new)symbol b ”inthe mannerofKing[ 115 ].Ofcourse,thiscouldbeconvenientlyextendedsothatast atementlike “ (1 ) = 2:=7 ”means“denethesymbol sothat (1 ) = 2=7 holds”[ 114 ].Wewill employthetypicalsymbol S c todenotethecomplementoftheset S ;furtherlet A [ B denotethe set-difference, A \ B c .Agreetolettheexpression x y meanthatthevalueofthevariable y isassignedtothevariable x .Todenotethecardinalityofaset S ,weuse j S j .Finally,wewilluse italics foremphasis,andSMALLCAPSforformalproblemnames.Anyotherlocallyusedterms andsymbolswillbedenedinthesectionsinwhichtheyappea r. 2.3FundamentalResults Inglobaloptimization,theobjectiveistodeterminethema ximumorminimumpoint attainedbyanobjectivefunctiondenedoveraset.Ingener al,anoptimizationproblemhasthe form minimizeormaximize f ( x ) subjectto x 2 S; where S R n isthefeasibleregionand f ( x ) isarealvaluedfunctiondenedon S .Thatis, f : S 7! R 1 Denition1. Anoptimizationproblemwithfeasibleregion S R n issaidtobe infeasible if S = ; Throughoutthisdissertation,wewillrelyheavilyontheno tionofa neighborhood whichis denednext. 17


Denition2. Foragivenoptimizationproblemonaset S R n ,a neighborhood isamapping N : S 7! 2 S denedforeachinstance. Insubsequentchapterswewillseethatcleverlydeningane ighborhoodforaparticular problemcangreatlyincreasetheeffectivenessofheuristi cs.Forexample,if S = R n ,thentheset ofpointsthatfallwithinsomeEuclideandistanceprovidea naturalchoicefortheneighborhood [ 144 ].If jjjj representstheEuclideannorm,thenapoint x 2 S issaidtobea localminimum pointof f if f ( x ) f ( x ) forallpoints x 2 S suchthat jj x x jj ,forsome 0 : Inother words,given > 0 ,denetheneighborhoodof x as N ( x ):= f x : x 2 S and jj x x jj g : (2–2) Then, x isalocalminimumif f ( x ) f ( x ) forall x 2 N ( x ) .Apoint x issaidtobea global minimum if f ( x ) 0 ,there exists > 0 suchthatif x 2 I and j x c j < ,then j f ( x ) f ( c ) j < Denition4. Aset S R n issaidtobe closed ifScontainsthelimitsofallconvergent sequencesofpoints x i 2 S: Denition5. Aset S R n issaidtobe compact ifSisbothclosedandbounded. Weknowfromrealanalysisthatif S isacompactset,theneveryinnitesequenceofpoints x i 2 S hasaconvergentsubsequencewhoselimitisin S [ 161 ].Furthermore,foracontinuous 18


function f f ( x i ) ( i !1 ) f ( x ) whenever x i ( i !1 ) x .Thisleadsustofundamentalresultby Weierstrasswhichwestatewithoutproof[ 107 ]. Theorem1. If S isanonemptycompactsetin R n ,and f ( x ) isacontinuousfunctiondenedon S ,then f ( x ) hasatleastoneglobalminimum(maximum)pointin S Wecannowmoveonandexaminesomepropertiesoflocalandglo balminima.Recallfrom calculusthatifthefunction f iscontinuouslydifferentiableinaneighborhoodofapoint x 2 S and d 2 R n ,then d T r f ( x ) issaidtobethe directionalderivative of f at x inthedirection d Ifwex x and d ,thenthefunction h ( ):= f ( x + d ) ,for 2 R + ,describes f alongtheray f x = x + d; 0 g : Ifweevaluatethederivativeof h withrespectto atthepoint =0 usingtherstorderTaylorexpansionof h at ,weseethatthisisprecisely d T r f ( x ) .Thus, d T r f ( x ) < 0 impliesthatthereexists > 0 suchthat f ( x + d ) 0 suchthat x + d 2 S forevery 0 < Thisleadsustothefollowingtheoremwhichprovidesaneces saryconditionforlocally optimalsolutions.Theorem2. Supposethatthefunction f ( x ) iscontinuouslydifferentiableonanopenset containing S R n .If x isalocalminimumof f (withrespectto S ),then d T r f ( x ) 0 for every d 2 Z ( x ) ,where Z ( x ) isthesetofallfeasibledirectionsat x x 2 S iscalleda criticalpoint if d T r f ( x ) 0 forevery d 2 Z ( x ) .Now,wewill determineinwhichcasescriticalpointsrepresentglobalo ptima.Werstneedtorecallthe denitionsofconvexsetsandconvexfunctions.Denition7. Aset S R iscalleda convexset ifforevery x 1 ;x 2 2 S ,and 2 R ; 0 1 thepoint x 1 +(1 ) x 2 2 S 19


Inageometricsense, S isconvexifforanytwopointsin S ,thelinesegmentjoiningthese twopointsiswhollycontainedin S [ 148 ]. Denition8. Givenaconvexset S R n ,thefunction f : S 7! R issaidtobea convexfunction ifforany x 1 ;x 2 2 S ,and 2 R ; 0 1 ,thefollowingconditionholds: f ( x 1 +(1 ) x 2 ) f ( x 1 )+(1 ) f ( x 2 ) : Thefunction f issaidtobea concavefunction ifandonlyif f isconvex.Inthefollowing theorem,weprovethatforoptimizationproblemswhere f isconvexand S isaconvexset,that criticalpointsarealwaysgloballyoptimalsolutions.Theorem3. Let f : S 7! R beaconvexfunction,where S R n ,isaconvexset.Thenevery localminimumof f isalsoaglobalminimum. Proof. Let x bealocalminimumpointandassumeforthesakeofcontradict ionthatthereexists anotherpoint x 2 S suchthat f ( x ) 0 suchthatfor 0 < f ( x + ( x x ) f ( x ) : Inthefollowingtheorem,weprovideanoptimalitycriterio nforminimizingaconcave functionoveraconvexset.Theorem4. Givenacompactconvexset S R n ,theglobalminimumofaconcavefunction f : S R isattainedatanextremepointof S Proof. Weknowthatanypoint x inaconvexsetcanbewrittenastheconvexcombinationofthe extremepoints v i of S .Thatis, x = P Ni =1 i v i ,suchthat P Ni =1 i =1 ; i 0 8 i: Since f is 20


concave,wehave f ( x ) N X i =1 i f ( v i ) N X i =1 i min f f ( v i ): i =1 ;:::;N g (2–3) =min f f ( v i ): i =1 ;:::;N g : (2–4) Recallthatlinearfunctionsarebothconvexandconcave.Th ereforeifweareconsidering a linearprogrammingproblem ,i.e.thatofminimizingacontinuouslinearfunctionovera polytope,bothTheorem 3 providingoptimalityoflocalminimaforconvexfunctionsa nd Theorem 4 providingextremepointoptimalityofconcavefunctionsap ply.Thus,forthisclassof problemswecanrestrictthesearchfortheglobalsolutionb yexaminingonlytheextremepoints ofthepolytope.Intheabsenceofconvexityhowever,agloba lminimumpointcanoccurata pointotherthananextremepoint. Inthisdissertation,wefocusonproblemsofthistype.Inpa rticulartheproblemswewill laterinvestigatecontainmanylocallyoptimalsolutionsw hichdifferfromtheglobalsolution. Also,untilnowwehavefocusedontheoremsforcontinuousfu nctions.However,several problemswewillencounterarehavediscretevariables.The seproblemsarecalled combinatorial optimizationproblems .Thenextsectioncontainssomebasicresultsregardingcom binatorial problemswhichwewilllateruse. 2.4DiscreteOptimization Incertainapplications,itisnecessarytorestricttheval uesofthedecisionvariablesofa problemtobeintegervalued.Suchproblemsarereferredtoa s integerprogrammingproblems Sometimes,itisconvenienttoincludeintegervariablesin aproblemwhenoneisattemptingto modelasituationthathastwopossiblevalues.Inthiscase, binaryvariablesthattakethevalue 0 or 1 areused. Integerprogrammingproblemspresentuniquechallengesin thatthetechniquesand theoremsforlinearprogrammingproblemsasdescribedabov edonotnecessarilyapply.For example,considerthepolytopeinFigure 2-1 .Noticethattheintegerpointsdonotlieatthe 21


Figure2-1:Noticethattheroundedintegersolutionisnoto ptimal. extremepointsofthepolytope.Weseethenthattheresultfr omTheorem 4 doesnothold. Anothercommonmisconceptionisthattheintegeroptimalso lutioncanbefoundbyrounding thelinearprogrammingsolutiontothenearestinteger.Wes eebyexaminingthegurethatthis doesn'tworkeither.Noticethattheintegerpointnearestt helinearprogrammingoptimalsolution fallsoutsidethefeasibleregionofthepolytope. Clearly,weneedmoreadvancedmethodsforsolvingsuchprob lems.Wewilllookat avarietyofexactandheuristicmethodsinSection 2.7 .Now,weprovideanintroductionto computationalcomplexity.Complexitytheoryliesatthehe artofglobaloptimizationand providestoolsforempiricallydeterminingthelevelofdif cultyofagivenproblemaswellasthe effectivenessofanalgorithm.Laterwewillconrmoursusp icionsaboutthedifcultyofinteger programsandnonconvexcontinuousprogrammingproblems. 2.5ComputationalComplexity Inthissection,wedevelopameansbywhichwecanclassifyap roblemaseitherbeing “easy”or“hard”.Thenfortheso-called“hard”problems,we lookatwaystoanswerthe question: “howhardishard?” Analgorithmissaidtobea polynomialtimealgorithm ifitsnumberofelementary operations,i.e.itsrunningtimeonacomputer,areinthewo rstcast,boundedabovebya 22


Figure2-2:Visualizationofcomplexityclasses. polynomialinthesizeoftheinput[ 107 ].Forinstance,analgorithmissaidtobe O ( I p ) ifthe polynomialwhichboundstherunningtimeisoforder p inthesizeoftheinputdata I .An algorithmissaidtobean exponential timealgorithmifitisnotboundedbyapolynomialinthe lengthoftheinput[ 6 ]. WhendiscussingproblemsinOR,wesplitthecollectionofal lproblemsintotwoclassesas visualizedinFigure 2-2 .Thoseproblemswhichcanbesolvedoptimallybyapolynomia ltime algorithmaresaidtobelongtotheclass P .Theothercomplexityclasscontainsthoseproblems whichcanbesolvedbya“nondeterministic”algorithminpol ynomialtime.Thisclassiscalled NP .Aproblemin NP isoneinwhichitiseasytoverifythecorrectnessofsolutio n,butisvery hardtosolve,whereaproblembelongingto P issimply“easy”tosolve. Amongtheproblemsin NP ,thosewhicharethemostdifculttosolvearesaidtobe NP -complete.Aproblem 1 issaidtobepolynomiallytransformabletoproblem 2 ifa polynomialtimealgorithmfor 2 wouldimplyapolynomialtimealgorithmfor 1 .Problems in NP -completearespecialinthateveryproblemin NP canbepolynomiallytransformedto everyotherproblemin NP -complete.Thus,since PNP ,itfollowsthatifonecoulddesign apolynomialtimealgorithmforasingle NP -completeproblem,theneveryproblemin NP couldbesolvedwithapolynomialtimealgorithmandthus P wouldequal NP [ 79 ].However, 23


Table2-1:Growthratesofseveralpolynomialandnon-polyn omialfunctions. n n 1 n 2 n 4 2 n n 10 10 1 10 2 10 4 10 3 3 : 6 10 6 100 10 2 10 4 10 8 1 : 27 10 30 9 : 33 10 157 1000 10 3 10 6 10 12 1 : 07 10 301 4 : 02 10 2 ; 567 10 ; 000 10 4 10 8 10 16 0 : 99 10 3 ; 010 2 : 85 10 35 ; 659 despitetheincredibleamountofresearchandinvestigatio n,thequestionastowhether P = NP remainsthesinglegreatestunsolvedproblemintheoretica lcomputerscience[ 145 ].Infact,the ClayMathematicsInstitutehasnamedthisproblemasoneofs evenprizemillenniumproblems andisoffering$1milliontoanyonewhopresentsananswerto thequestion,“does P = NP ?” [ 109 ]. Denition9. Anoptimizationproblem issaidtobe NP -hard,ifthereexistsand NP -complete problemwhichispolynomiallytransformableto Throughoutthisdissertation,wearegoingtofocusonprobl emsthatare NP -hardand NP -complete.Thenextreasonablequestiononeasksof NP problemsiswhattheimplicationis forsolvingthem.Thatis,howdoesbeingin NP reallycomplicatethecomputationaltractability ofaproblem.Table 2-1 providesaseveralexamplesofthegrowthratesofsomepolyn omial andexponentialfunctions[ 6 ].Noticehowquicklytheexponentialalgorithmsgrow.This isone reasonwhypolynomialtimealgorithmsarepreferredoverex ponentialtimealgorithms.Most discreteoptimizationproblemsturnouttobeeither NP -hardor NP -complete,eveniftheyare linear[ 149 ]. 2.6UpperandLowerBounds Whenattemptingtosolveintegerprograms(IPs),weareface dwiththeproblemofhowto provethatagivenpointisanoptimalsolution[ 173 ].Thisproblemarisessincelocaloptimality doesnotimplyglobaloptimalityforIPs.Oftentimesbeinga bletoderiveupperandlowerbounds ontheoptimalsolutionishelpfultoidentifygoodapproxim atesolutionsandnarrowthesearch fortheoptimalsolution.Thistopicwillbestudiedextensi velyinChapter 5 .Nowweintroduce somebasicpropertiesofboundsforintegerprogrammingpro blems. 24


ConsidertheIPgivenbelowandassumethatthepoint x isanoptimalsolution. minimize cx (2–5) subjectto x 2 S (2–6) x 2 Z n : (2–7) TheninordertosolvethisIP,weneedtodeterminealowerbou nd x suchthat c ( x ) c ( x ) and anupperbound x where c ( x ) c ( x ) suchthat c ( x )= c ( x )= c ( x ) : (2–8) Inorderndtheseboundsinpractice,weneedanalgorithmth atcancomputeadecreasing sequenceofupperbounds c ( x 1 ) >c ( x 2 ) >:::>c ( x s ) c ( x ) ; (2–9) andacorrespondingincreasingsequenceoflowerbounds c ( x 1 ) 0 [ 173 ]. Weusetheideaofarelaxationinordertondthesebounds.Ar elaxationtypicallyenlarges thesetoffeasiblesolutions,butiseasiertosolvethanthe originalproblem. Denition10. Aproblem(RP) z R =max f f ( x ): x 2 T R n g issaidtobea relaxation of(IP) z =max f c ( x ): x 2 S R n g if: (i) S T ,and (ii) f ( x ) c ( x ) forall x 2 S Usingthisdenition,thefollowinglemmaholds. Lemma1. If(RP)isarelaxationof(IP),then z R z 25


Proof. Let x 2 S beanoptimalsolutionfor(IP).Then,wehave x 2 S T ,which implies c ( x ) f ( x ) .Furthermore,since x 2 T f ( x ) isalowerboundon z R .Thatis, z = c ( x ) f ( x ) z R ; andwehavethelemma. Theproblemofformulatingusefulrelaxationsisanimporta ntprobleminitsownright whichhasbeenstudiedsincethefoundingofOR[ 53 ].Amongthemostcommonrelaxations arethelinearprogrammingrelaxation,andtheLagrangianr elaxation.Wenowprovideabrief introductiontothese.Denition11. Givenanintegerprogram(IP) z :=max f cx : x 2 S \ Z n R n g ,the linear programmingrelaxation of(IP)isthelinearprogram(LPR) z LP :=max f cx : x 2 S R n g Proposition1. Thelinearprogrammingproblem(LPR)isarelaxationof(IP) Proof. Theproofistrivialas S \ Z n S andtheobjectivefunctionof(LPR)remainsthesame asin(IP).Thus,byLemma 1 ,wehavetheresult. Weseethenthatalllinearprogrammingrelaxationsprovide boundsontheoriginalinteger program.Further,thefollowinglemmashowsthatrelaxatio nscanbehelpfulforidentifyingcases inwhichtheoriginalintegerprogramisinfeasible[ 173 ]. Lemma2. Givenanintegerprogram(IP) z IP :=max f cx : x 2 S \ Z n R n g ,andits correspondingLPrelaxation(LPR) z LP :=max f cx : x 2 S R n g ,thefollowingstatements hold. (i) If(LPR)isinfeasible,then(IP)isalsoinfeasible. (ii) If x isanoptimalsolutionto(LPR)suchthat x 2 S \ Z n and f ( x )= c ( x ) ,then x is anoptimalsolutionfor(IP). Proof. (i)Thisfollowsfromthefactthat S \ Z n S .Since(LPR)isinfeasible,wehavethat S = ; ,thusimplying S \ Z n = ; (ii)Since(LPR)isarelaxation,thenbyDenition 10 x 2 S \ Z n implies z IP c ( x )= f ( x )= z LP : However,bydenition z IP z LP implying c ( x )= z IP = z LP 26


Anothercommonrelaxationusedtotacklehardintegerprogr amsistheLagrangian relaxation.ThismethodwasrstintroducedbyHeldandKarp in[ 96 97 ]inaformulationfortheTRAVELINGSALESMANPROBLEM[ 54 ].Lagrangianrelaxationrelaxestheconstraintsbyadding themtotheobjectivefunctionwithanassociatedpenalty.C onsiderthefollowingoptimization problem, (P) z =max cx (2–12) s.t. Ax b (2–13) x 2 X: (2–14) ThenthecorrespondingLagrangianrelaxationisformulate dasfollows. P( ) L ( )=max cx + ( b Ax ) (2–15) x 2 X: (2–16) WiththiswecanformulatethefollowinglemmaregardingLag rangianrelaxation. Lemma3. P ( ) isarelaxationof ( P ) Proof. Inorderfor P( ) tobearelaxationof(P),asdenedbyDenition 10 ,wemustshow thefollowingtwoconditions:(i)thefeasibleregionofthe originalproblemisasubsetofthe relaxedproblem,and(ii)forallvectors 0 L ( ) z Condition(i)followstrivially.Toshowcondition(ii),le t x 2 X beanoptimalsolution for(P).Then x isclearlyfeasiblefor P ( ) .Further,since x feasiblefor(P), b Ax 0 Therefore cx cx + ( b Ax ) ,forallrealvectors 0 Fromthislemma,weseethatanyfeasiblesolutionto(P( ))isanupperboundonthe optimalvalueof(P).Theproblemofndingthebest,or tightest bound,say L isknownasthe LAGRANGIANMULTIPLIERPROBLEM[ 124 ]andisgivenas L :=min f L ( ): 0 g .With thisthefollowinglemma,whichisstatedwithoutproof,hol ds. 27


Lemma4. Forallrealvectors cx z L L ( ) .Furthermore,if L ( )= L = z = cx ,then x isoptimalfortheoriginalproblem(P),and isoptimalforthe LAGRANGIAN MULTIPLIERPROBLEM. LinearprogrammingandLagrangianrelaxationsarehelpful forshowingwhenasolutionis closeto(insomecasesequals)theoptimalsolution.Theyar ealsousefulinbranchandbound algorithms,whichwewillintroduceinthefollowingsubsec tion. 2.7AlgorithmsforOptimizationProblems Consideradiscreteoptimizationproblemthathas n binarydecisionvariables.Sincethere areanitenumberofintegerfeasiblesolutions,intheoryo necouldenumerateallpossible solutions.However,todothiswouldrequire 2 n functionevaluations[ 31 ].Thisisimpractical sinceif n> 1000 ,thenwiththepresentcomputersavailablethecomputation timerequiredfor thisenumerationwouldtakemillionsofyears.Clearlywene edmoreefcientalgorithmstosolve theseproblems. Algorithmsforoptimizationproblemsarebrokenupintotwo categories:exactmethods andheuristics.Exactmethodsguaranteethattheterminati onofthealgorithmwillresultin theoptimalsolutionprovidedoneexists.Heuristicsonthe otherhand,havenoguaranteeof optimalitybutusuallyndhighqualitysolutionsmuchfast erthantheexactmethods.Infact, mostnontrivialinstancesofproblemsin NP cannotbesolvedbyexactalgorithms.Thus,we needefcientheuristicstondnearoptimalsolutionstore al-worldinstances.Inthefollowing subsections,weprovideanoverviewofseveralexactandheu risticmethodsthatwewillusefor solvingtheproblemsappearinginlaterchapters.2.7.1ExactMethods LinearProgrammingTechniques .Webeginourdiscussionofexactalgorithmswiththe simplexmethod forlinearprogramming[ 52 ].Consideraninstanceofalinearprogramming 28


problem minimize cx (2–17) subjectto Ax = b (2–18) x 0 ; (2–19) where A isan m n matrixwithrank m c an n -vector,and b T isan m -vector.Thesimplex methodisanalgorithmwhichmovesalongtheextremepointso fthepolytopedenedby Ax = b insearchoftheoptimalsolution.Theextremepointsarevis itedinsuchawaythatthe objectivefunctionvalueatanewpointisatleastasgoodast heprevious.Sinceweshowedin theprevioustheoremsthattheoptimalsolutiontoalinearp rogramisanextremepointofthe polytope,thesimplexmethodisguaranteedtondtheoptima lsolution.Noticehowever,that thisalgorithmisnotpolynomial.Thepolytopeinquestionc anhave n m extremepoints,andit ispossibletoconstructexamplesforwhichthesimplexmeth odmustenumerateall n m ofthem. However,despitethetheoreticalexponentialworst-casec omplexityofthesimplexmethod,it isveryefcientinpracticeandiseasytoimplement.Thisis nottosaythatlinearprogramsare NP hard .Infact,alllinearprogramsarein P .Theclassofalgorithmsknownas interiorpoint methods areabletosolvelinearprogrammingproblemsinpolynomial time.Therstefcient interiorpointalgorithmwasproposedin1984byKarmarkar[ 111 ].Therstimplementationof Karmarkar'salgorithmwasreportedin1991byAdler,Karmar kar,Resende,andVeigain[ 4 ] and[ 5 ].Excellentreferencetextsonlinearprogramminginclude theworkofChvatal[ 34 ]and Bazaraaetal.[ 15 ]. IntegerProgrammingAlgorithms .Branchandbound(B&B)algorithmsarethemost commonclassofalgorithmsusedforsolvingdiscreteoptimi zationproblems[ 61 ].B&Bmethods are implicitenumeration techniquesbasedontheideaof divideandconquer [ 90 ].InaB&B algorithm,thesetoffeasiblesolutionsisdecomposedinto smallerandsmallersetsuntilthe optimalsolutioniseventuallyreached. 29


Considertheintegerprogrammingproblem z IP :=max f cx : x 2 S \ Z n R n g .To applytheB&Balgorithm,thelinearprogrammingrelaxation z LP :=max f cx : x 2 S g issolved, generallyresultinginanonintegralsolution.Thissoluti onistakenasinitialupperboundonthe optimalsolution.SupposethatintheLPrelaxation,someva riable x i = x i 62 Z .Thenonewayto branchistodividethefeasibleregion S intotwosubdomains,namely S 1 := S \f x : x i b x i cg (2–20) S 2 := S \f x : x i d x i eg (2–21) Noticethat S 1 [ S 2 = ; and S 1 \ S 2 = ; .Thetwolinearprograms z LP 1 :=max f cx : x 2 S 1 g and z LP 2 :=max f cx : x 2 S 2 g arenowsolvedandthesmallestobjectivevalueistakenasth enew upperbound.Inessence,a searchtree isformedbytherepetitionofthedecomposition/bounding processappliedtoeachsub-problem.However,duetoapre-e stablishedlowerbound,manyof theresultingsubproblemsare pruned fromthesearchtreeandnotconsidered.Thus,anoptimal permutationisconstructediteratively,oneelementatati me[ 151 ].Theprocessisrepeatedon variableswhicharenonintegraluntileventuallytheinteg eroptimalsolution z IP isreached[ 173 ]. Thoughbranchandboundmethodsarethemostcommonlyusedal gorithmsfordiscrete optimizationproblems,theyarenottheonlytechniquesava ilable.Thebranchandcutmethod isahybridofbranchandboundwhichfallsintotheclassofso -called cuttingplane techniques [ 54 107 173 ].IntroducedbyGomoryin[ 88 ],cuttingplanemethodssolveIPsbyintroducing constraintswhichcutsoffthenonintegersolutionfoundby solvingtheLPrelaxationwithout removinganyfeasibleintegersolutions.Finally,columng eneration,orso-calledbranchandprice algorithmsareeffectivedecompositionmethodsandarecom monlyusedforsolvinglarge-scale integerprogrammingproblems[ 55 57 58 ].Thedesignofsoftwarepackageswhichefciently executeoptimalintegerprogrammingalgorithmsisaglobal enterprize.Todaythemostefcient andwidelyusedcommercialIPsolversareCPLEX R r byILOG,Inc.[ 50 ]andXpress-MP R r by DashOptimizationInc.[ 108 ].Inlaterchapters,wewillapplybranchandboundtechniqu esto severalproblemsinordertondtheoptimalsolutionsandve rifytheeffectivenessofheuristics. 30


2.7.2Heuristics Despitetheguaranteeofeventuallyreachingtheoptimalso lution,B&Bmethodsare inefcientonlargeproblems.Therefore,wemustlookforef cientwaysofproducinghigh qualitysolutions.Heuristics,orsuboptimalalgorithmsp rovidethisoutlet.Thetermheuristicis derivedfromtheGreekword heuriskein ( ),meaning“tondordiscover”.Heuristics areapproximationalgorithmsandaretheonlyalternativet onding“good”feasiblesolutions whenproblemsaretoodifculttoapplybranchandboundmeth ods.Thestudyofheuristicsis vast,andhaslettothecreationofalgorithmicmethodswhic harecapableofproducingexcellent solutionsinseconds,forproblemsinwhichaB&Borotheropt imalalgorithmwouldrequire yearstosolve. Inthefollowingparagraphs,weprovideabriefintroductio ntoseveralheuristicswhichwe willanalyzeinlaterchaptersofthisdissertation.Webegi nwithathesimplesttypeofheuristic knownasthe greedyalgorithm GreedyHeuristicsandLocalSearch .Agreedyalgorithmisalocalsearchmetaheuristic whichgetsitsnamefromthemyopicwayinwhichitcreatescan didatesolutions[ 123 ].Ateach step,thegreedymethodmakeswhateverchoiceseemsbestatt hatparticularmomentintime. Onceadecisionismade,itispermanentandcannotbelaterch anged.Therefore,onemustensure thatacandidateelementisfeasiblebeforeaddingittothei ncumbentsolution[ 105 ]. Anexampleofagreedyalgorithmisasfollows.Supposeacash ierowesacustomer $0.42cents.Thecashiercanusethegreedymethodtodetermi netheminimumnumberof coinsrequiredforthistransaction.Pseudo-codeforthism ethodisprovidedinFigure 2-3 .The algorithmtakesasinput n ,theamountofchangedue,inthiscase, n =$0 : 42 .Tobeginwith, onequarterisselectedbringingthebalanceto$0.17.Next, onedimeischosenandtheremainder is$0.07.Byselectingonenickelandtwopennies,theproble missolved.Weseethatgreed ismanifestedinthisexampleasthealgorithmselectsthehi ghestvaluedcoinsrst.Forthis problem,thegreedyalgorithmcomputestheoptimalsolutio nfromthe 31 uniquecombinationsof coinswhichaddupto$0.42cents. 31


procedure GreedyChangeMaker ( n ) 1 C f 1 ; 5 ; 10 ; 25 g 2 Change ; 3 Sum 0 / sumofcoinsin Change / 4 while Sum 6 = n do 5 x max f c 2 C : Sum + c n g 6 if 69 x then 7 return:NOSOLUTION 8 else 9 Change Change [ x 10 sum sum + x 11 endif 12 endwhile 13 return Change endprocedure GreedyChangeMaker Figure2-3:Pseudo-codeforagreedyalgorithmwhichmakesc hangeusingtheminimumnumber ofcoins. Otherproblemsforwhichthegreedymethodndstheoptimals olutionincludetheMINIMUMSPANNINGTREEproblemwhereKruskal'salgorithm[ 122 ]ndsaminimum weightspanningtreeofagivengraph[ 6 ].Despitetheperformanceofthegreedyalgorithm ontheaboveexample,greedymethodsalmostalwaysfallshor toftheoptimalsolutionwhen appliedto NP -completeproblems.Thisisbecausegreedymethodsselecta localoptimumfrom theneighborhoodofthecurrentsolutionateachstepwithth ehopethatintheend,theglobal optimumisfound.Howeveraswelearnedearlierinthechapte r,thisisn'tnecessarilythecase. Otherlocalsearchheuristicsinvolvesimpleexaminations ofneighborhoodsinthequest fora“good”solution[ 51 ].Themethodmovesfromonesolutiontothenextinthefeasib le region,untilthecurrentsolutioncannotbeimprovedbysel ectinganalternatesolutioninits neighborhood.Thespecicneighborhoodstructuredepends upontheproblemathand,and asmentionedearlier,acleverchoiceofneighborhoodcangr eatlyimprovetheefcacyofthe heuristic.Popularlocalsearchmethodsincludethe2-exch ange(or,2-opt)method[ 173 ],hill climbingprocedures,themethodofconjugatedgradients[ 91 92 ],andsteepestascent/descent methods.Wewillseeseveralexamplesoflocalsearchalgori thminthelaterchapters.For 32


procedure GRASP ( MaxIter ; RandomSeed ) 1 f 0 2 X ; 3 for i =1 to MaxIter do 4 X ConstructionSolution ( G;g;X; ) 5 X LocalSearch ( X;N ( X )) 6 if f ( X ) f ( X ) then 7 X X 8 f f ( X ) 9 end 10 end 11 return X endprocedure GRASP Figure2-4:GRASPformaximization detailedimplementationspecications,oneshouldconsul ttextbookonlocalsearch,suchasthe workof[ 2 ].Foranannotatedbibliographyoflocalsearch,thereader isalsoreferredto[ 3 ]. GreedyRandomizedAdaptiveSearchProcedure(GRASP) GRASP[ 69 ]isamulti-startmetaheuristicthathasbeenusedwithgrea tsuccessto providesolutionsforseveraldifcultcombinatorialopti mizationproblems[ 72 ],includingSATISFIABILITY[ 154 ],JOBSHOPSCHEDULING[ 7 ],VEHICLEROUTING[ 32 ],andQUADRATIC ASSIGNMENT[ 128 140 ].ForanannotatedbibliographyofGRASP,thereadershould reference thepaperbyFestaandResende[ 72 ]. GRASPisatwo-phaseprocedurewhichgeneratessolutionsth roughthecontrolleduseof randomsampling,greedyselection,andlocalsearch.Forag ivenproblem ,let F bethesetof feasiblesolutionsfor .Eachsolution X 2 F iscomposedof k discretecomponents a 1 ;:::;a k GRASPconstructsasequence f X g i ofsolutionsfor ,suchthateach X i 2 F .Thealgorithm returnsthebestsolutionfoundafteralliterations.TheGR ASPprocedurecanbedescribedasin thealgorithmpresentedinFigure 2-4 .The constructionphase receivesasparametersaninstance oftheproblem G ,arankingfunction g : A ( X ) 7! R (where A ( X ) isthedomainoffeasible components a 1 ;:::;a k forapartialsolution X ),andaparameter 0 << 1 .Theconstruction phasebeginswithanemptypartialsolution X .Assumingthat j A ( X ) j = k ,thealgorithmcreates 33


alistofthebestranked k componentsin A ( X ) ,andreturnsauniformlychosenelement x from thislist.Thecurrentpartialsolutionisaugmentedtoincl ude x ,andtheprocedureisrepeated untilthesolutionisfeasible,i.e.until X 2 F The intensicationphase consistsoftheimplementationofahill-climbingprocedur e.Given asolution X 2 F ,let N ( X ) bethesetofsolutionsthatcanfoundfrom X bychangingoneof thecomponents a 2 X .Recallthat N ( X ) iscalledtheneighborhoodof X .Theimprovement algorithmconsistsofnding,ateachstep,theelement X suchthat X :=argmax X 0 2 N ( X ) f ( X 0 ) ; where f : F 7! R istheobjectivefunctionoftheproblem.Attheendofeachst epwemake X X if f ( X ) >f ( X ) .Thealgorithmwilleventuallyachievealocaloptimum,inw hich casethesolution X issuchthat f ( X ) f ( X 0 ) forall X 0 2 N ( X ) X isreturnedasthe bestsolutionfromtheiterationandthebestsolutionfroma lliterationsisreturnedastheoverall GRASPsolution. SimulatedAnnealing .Instatisticalmechanics,thephysicalprocessofanneali ngisusedto relaxasystemtothestateofminimalenergy.Thisisdonebyh eatingthesoliduntilitmeltsand thencoolingitslowlysothatateachtemperaturethepartic lesrandomlyarrangethemselvesuntil reachingthermalequilibrium. In[ 116 ],Kirkpatricketal.introducedamethodforcombinatorial problemsknownas simulatedannealing .Basedonthetheoryofthephysicalprocess,simulatedanne alingwas showntoasymptoticallyconvergetotheglobaloptimumafte rperforminganumberofso-called transitionsatdecreasingtemperatures. Pseudo-codeforagenericsimulatedannealingalgorithmis presentedinFigure 2-5 .The algorithmtakesasinputtheinitialtemperature T andareductionfactor r 2 (0 ; 1) .Simulated annealingessentiallychoosesaneighboratrandomtorepla cetheincumbentsolution.Ifthe chosenneighborisabettersolutionthenitisacceptedwith probability 1 .However,inorderto escapeandevadelocaloptima,ifthechosenneighboriswors ethantheincumbent,thenitis 34


procedure SimulatedAnnealing ( T;r ) 1 f 0 2 X ; 3 X randomSolution () 4 while T 6 =0 do 5 for i =1 to MaxIter do 6 X 0 randomNeighbor ( X ) 7 if f ( X 0 ) f ( X ) then 8 X X 0 9 else 10 X X 0 withprobability e f ( X 0 ) f ( X ) T 11 endif 12 T rT 13 if f ( X ) >f then 14 X X 15 f f ( X ) 16 endif 17 endfor 18 endwhile 19 return X endprocedure SimulatedAnnealing Figure2-5:Genericsimulatedannealingmaximizationalgo rithm. acceptedwithsomepositiveprobabilitywhichisadecreasi ngfunctionofthetemperature[ 1 ]. Thusthecoolingschedule,orthemethodinwhichthetempera turedecreasesisanimportant partoftheheuristic.Ithasbeenshownthatalogarithmical lyslowcoolingscheduleguarantees thatthealgorithmwillconvergetotheglobaloptimuminexp onentialtime[ 24 ].Thereforein practice,fastercoolingschedulesareoftenused.Another methodcloselyresemblingsimulated annealingisthemethodofmeaneldannealing. Meaneldannealing(MFA),isaheuristicwhichmimicstheid eaofmeaneld approximationfromstatisticalphysics[ 150 ].InMFA,thestochasticprocessinsimulated annealingisreplacedbyasetofdeterministicequations.T houghMFAdoesnotguarantee convergencetoaglobaloptimalsolution,itcanprovideane xcellentapproximationtoanoptimal solutionandismuchlessexpensivecomputationally. GeneticAlgorithms 35


procedure GeneticAlgorithm 1 Generatepopulation P k 2 Evaluatepopulation P k 3 while terminatingconditionnotmet do 4 Selectindividualsfrom P k andcopyto P k +1 5 Crossoverindividualsfrom P k andputin P k +1 6 Mutateindividualsfrom P k andputin P k +1 7 Evaluatepopulation P k +1 8 P k P k +1 9 P k +1 ; 10 endwhile 11 return bestindividualin P k endprocedure GeneticAlgorithm Figure2-6:Pseudo-codeforgenericgeneticalgorithm. Geneticalgorithmsreceivetheirnamefromanexplanationo fthewaytheybehave.Itcomesas nosurprise,theyarebasedonDarwin'sTheoryofNaturalSel ection[ 56 ].Geneticalgorithms storeasetofsolutions,ora population ,andthepopulation evolves byreplacingthesesolutions withbetteronesbasedoncertaintnesscriterionrepresen tedbytheobjectivefunctionvalue. Insuccessiveiterations,or generations ,thepopulationevolvesby reproduction crossover and mutation .Reproductionistheprobabilisticselectionofthenextge nerationselements determinedbytheirtnesslevel.Crossoveristhecombinat ionoftwocurrentsolutions, called parents whichproducesoneormoreothersolutions,referredtoas offspring .Finally, mutationistherandommodicationoftheoffspring.Mutati onisperformedasanescape mechanismtoavoidgettingtrappedatalocaloptimum[ 86 ].Insuccessivegenerations,only thosesolutionshavingthebest tness arecarriedtothenextgenerationinaprocesswhich mimicsthefundamentalprincipleofnaturalselection, survivalofthettest [ 56 ].Figure 2-6 providespseudo-codeforastandardgeneticalgorithm.Gen eticalgorithmswereintroducedin 1977byHolland[ 102 ],andweregreatlyinvigoratedbytheworkofGoldbergin[ 86 ]. 2.8ConcludingRemarks Inthischapterwehaveprovidedabriefhistoryandintroduc tiontoglobaloptimization.The chapterisnotintendedtobeallinclusive;instead,thepur poseofitsinclusionisasfollows.First, 36


wehaveprovidedthefundamentalresultsandunderlyingthe orythatwewillusethroughout thisdissertation.Thisincludesthetheoryofcomputation alcomplexity,andashortoverviewof themostcommonsolutiontechniqueswewillencounterandap plytoseveralproblemsaswe progress.Secondly,wehaveprovidedseveraldenitions,l emmata,andtheoremsthatwewill referenceinthechapterstocome.Theintentistohaveaconc iselocationtowhichthereadercan refer.Also,presentingthemajortheoremsherewillpreven tredundancyaswewillnotre-state thetheoremsineachchaptertheyareapplied.Wewillnowmov eonandbegintheexamination ofseveralcombinatorialproblemsthatoccurinmilitaryte lecommunicationnetworks.We concludethischapterwithalistofreferencesontheory,al gorithms,andapplicationsofglobal andcombinatorialoptimization. Excellentreferencesonglobalandcombinatorialoptimiza tionincludetheworkofDuand Pardalos[ 63 64 65 ],FloudasandPardalos[ 74 76 ],HorstandPardalos[ 106 ],Horst,Pardalos, andThoai[ 107 ],Pardalos[ 146 ],PardalosandResende[ 147 ],PardalosandRosen[ 148 ],and Wolsey[ 173 ]tonameafew.Perhapsthemostinclusiveone-stopreferenc eisthemonumental workofFloudasandPardalosinthesixvolume EncyclopediaofOptimization [ 75 ]. Thelistofalgorithmsisalsonotintendedtobeexhaustive. Otherexactalgorithmsinclude dynamicprogramming[ 16 ]andouterapproximationmethods[ 107 ].Othereffectiveheuristics includetabusearch[ 81 82 83 ],scattersearch[ 80 ],hybridheuristicswhichcombineelements ofseveralmethods[ 22 71 159 ],andalgorithmsdesignedforthespecicproblemwhichexp loit thecombinatorialstructureoftheproblem[ 25 30 43 139 ].Otheralgorithmicreferencebooks includeAhujaetal.[ 6 ],FloudasandPardalos[ 73 ],Goldberg[ 86 ],Minieka[ 130 ],Osman[ 142 ], andOsmanetal.[ 143 ]. 37


CHAPTER3 JAMMINGCOMMUNICATION NETWORKSVIACRITICALNODEDETECTION 3.1Introduction Inthischapter,westudytwovariantsoftheCRITICALNODEPROBLEM.Ingeneral,the objectiveoftheCRITICALNODEPROBLEM(CNP)istondasetof k nodesinagraphwhose deletionresultsinthemaximumnetworkfragmentation.Byt hiswemean,maximizethenumber ofcomponentsinthe k -vertexdeletedsubgraph.Studiescarriedoutinthislinei ncludethose byBavelas[ 14 ]andFreeman[ 78 ]whichemphasizenodecentralityandprestige,bothofwhic h areusuallyfunctionsofanodesdegree.However,theylacke dapplicationstoproblemswhich emphasizednetworkfragmentationandconnectivity. WecanapplytheCNPtotheproblemofjammingwiredtelecommunicationnetworks by identifyingthecriticalnodesandsuppressingthecommuni cationonthesenodes.Thiswill resultinthemaximumnumberofdisconnectedcomponentswhi chareunabletocommunicate witheachother.TheCNPcanalsobeappliedtothestudyofcovertterroristnetworks ,wherea certainnumberofindividualshavetobeidentiedwhosedel etionwouldresultinthemaximum breakdownofcommunicationbetweenindividualsinthenetw ork[ 118 ].Likewiseinorderto stopthespreadingofavirusoveratelecommunicationnetwo rk,onecanidentifythecritical nodesofthegraphandtakethemofine. TheCNPalsondsapplicationsinnetworkimmunization[ 36 176 ]wheremassvaccination isanexpensiveprocessandonlyaspecicnumberofpeople,m odeledasnodesofagraph, canbevaccinated.Theimmunizednodescannotpropagatethe virusandthegoalistoidentify theindividualstobevaccinatedinordertoreducetheovera lltransmissibilityofthevirus. Thereareseveralvaccinationstrategiesintheliterature [ 36 176 ]offeringcontrolofepidemic outbreaks;however,noneoftheproposedareoptimalstrate gies.Thevaccinationstrategies suggestedemphasizethe centrality ofnodesasamajorfactorratherthan critical nodeswhose deletionwillmaximizedisconnectivityofthegraph.Delet ionofcentralnodesmaynotguarantee afragmentationofthenetworkorevendisconnectivity,inw hichcasediseasetransmission 38


cannotbeprevented.Ofcourse,owingtoitsdynamicstature ,therelationshipsbetweenpeople, representedbyedgesinthesocialnetworkaretransientand thereisaconstantrewiringbetween nodes,andalternaterelationshipscouldbeestablishedin thefuture.Theproposedcritical nodetechniquehelpsinamaximumpreventionofdiseasetran smissionoveraninstanceofthe dynamicnetwork. Beforeproceeding,wementiononenalareainwhichtheCRITICALNODEPROBLEMnds severalapplications,andthatisintheeldoftransportat ionengineering[ 66 ].Twoparticular examplesareasfollows.Ingeneral,fortransportationnet works,itisimportanttoidentifycritical nodesinordertoensuretheyoperatereliablyfortransport ingpeopleandgoodsthroughoutthe network.Further,inplanningforemergencyevacuations,i dentifyingthecriticalnodesofthe transportationnetworkiscrucial.Thereasonistwo-fold. First,knowledgeofthecriticalnodes willhelpinplanningtheallocationofresourcesduringthe evacuation.Secondly,intheaftermath ofadisastertheywillhelpinre-establishingcriticaltra fcroutes. Borgatti[ 21 ]hasstudiedasimilarproblem,focusingonnodedetectionr esultingin maximumnetworkdisconnectivity.Otherstudiesinthearea ofnodedetectionsuchascentrality [ 14 78 ]focusontheprominenceandreachabilitytoandfromthecen tralnodes.However,little emphasisisplacedontheimportanceoftheirroleinthenetw orkconnectivityanddiameter. Perhapsonereasonforthisisbecausealloftheaforementio nedreferencesreliedonsimulationto conducttheirstudies.Althoughthesimulationshavebeens uccessful,amathematicalformulation isessentialforprovidinginsightandhelpingtorevealsom eofthefundamentalpropertiesof theproblem[ 138 ].Inthenextsection,wepresentamathematicalmodelbased onintegerlinear programmingwhichprovidesoptimalsolutionsfortheCRITICALNODEPROBLEM. Weorganizethischapterbyrstformallydeningtheproble manddiscussingits computationalcomplexity.Next,weprovideanintegerprog ramming(IP)formulationfor thecorrespondingoptimizationproblem.InSection 3.3 weintroduceaheuristictoquickly providesolutionstolarge-scaleinstancesoftheproblem. Wepresentacomputationalstudyin Section 3.4 ,inwhichwecomparetheperformanceoftheheuristicagains ttheoptimalsolutions 39


whichweredeterminedusingacommercialsoftwarepackage. Someconcludingremarksare giveninSection 3.5 3.2ProblemFormulations Denoteagraph G =( V;E ) asapairconsistingofasetofvertices V ,andasetofedges E Allgraphsinthischapterareassumedtobeundirectedandun weighted.Forasubset W V ,let G ( W ) denotethesubgraphinducedby W on G .Asetofvertices I V iscalledan independent or stableset ifforevery i;j 2 I; ( i;j ) 62 E .Thatis,thegraph G ( I ) inducedby I isedgeless.An independentsetis maximal ifitisnotasubsetofanylargerindependentset( i.e. ,itismaximalby inclusion),and maximum iftherearenolargerindependentsetsinthegraph. 3.2.1CriticalNodeProblem Theformaldenitionoftheproblemisgivenby: C RITICAL N ODE P ROBLEM (CNP) INPUT:Anundirectedgraph G =( V;E ) andaninteger k OUTPUT: A =argmin P i;j 2 ( V n A ) u ij G ( V n A ) : j A j k; where u ij := 8>><>>: 1 ; if i and j areinthesamecomponentof G ( V n A ) 0 ; otherwise. Theobjectiveistondasubset A V ofnodessuchthat j A j k ,whosedeletionresults intheminimumvalueof P u ij intheedgeinducedsubgraph G ( V n A ) .Thisobjectivefunction resultsinaminimumcohesioninthenetwork,whilealsoensu ringaminimumdifferenceinthe sizesofthecomponents.Anillustrationisbestsuitedtoex plainthechoiceofobjectivefunction. Consideranarbitraryunweightedgraphwith 150 nodes.Accordingtoourobjective,itismore preferabletohaveapartitionwith 3 componentswitheach 50 nodesasopposedtoapartition with 5 componentswithonehaving 146 nodesandtherestofthemhavingasinglenode. ThisproblemissimilartoMINIMUMk -VERTEXSHARING[ 133 ],wheretheobjectiveis tominimizethenumberofnodesdeletedtoachievea k -waypartition.Hereweareconsidering thecomplementaryproblem,whereweknowthenumberofverti cestobedeletedandwetryto 40


maximizethenumberofcomponentsformedandimplicitlylim itthesizesofthecomponents. Borgatti[ 21 ]hasgivenacomprehensiveillustrationtofacilitatetheu nderstandingofthe objectivefunctionanditsnon-triviality. WenowprovethattherecognitionversionoftheCNPis NP -complete.Considerthe followingdecisionproblemfortheCNP: K-C RITICAL N ODE P ROBLEM (K-CNP) INPUT:Anundirectedgraph G =( V;E ) andaninteger k QUESTION:Doesthereexistazerocost K -waypartitionof G bydeleting k nodesorless? Theorem5. The K-CRITICALNODEPROBLEMis NP -complete. Proof. Toshowthis,wemustprovethat( 1 )K-CNP2NP ;( 2 )Some NP -completeproblem reducestoK-CNPinpolynomialtime. ( 1 )K-CNP2NP sincegivenanygraph G =( V;E ) ,wecanverifythevalidityof G in polynomialtime.Morespecically,bydeletinganysetofat most k nodes,wedetermine ifthethereisazero-cost K -waypartitionof G in O ( j E j + j V j ) timeusingadepth-rst search[ 6 ]. ( 2 )Tocompletetheproof,weshowareductionfromtheK-INDEPENDENTSETPROBLEM(K-ISP)[ 24 ],whichiswell-knowntobe NP -complete[ 79 ].Recallthattheobjectiveof theK-ISPistodetermineif G containsanindependentsetcontainingatleast K nodes. Let G =( V;E ) beagraphinwhichweseekanindependentset.Therearenonec essary transformationsrequiredforthegraphinwhichwearesolvi ngthecorrespondingK-CNP. Wewillshowthata`yes'instanceoftheK-ISPcorrespondstoa`yes'instanceofthe K-CNPon G .Inparticular, G hasanindependentsetofsize K ifandonlyiftheK-CNPhasazerocostsolutionwhere k j V j K .Suppose G containsanindependentset I where j I j = K .NoticethattheobjectiveoftheK-CNPwillbe 0 asthesubgraphinduced bydeletingthenodesin V n I isedgeless.Therefore,a`yes'instanceoftheK-ISPimplies a`yes'instancefortheK-CNPwith k = j V j K 41


Toprovetheconverse,observethatthecostofanyK-CNPisatleast 0 .Thus,a`yes' instanceoftheK-CNPwouldimplythatoncethe k criticalnodesareremoved,theresulting subgraphconsistsof K componentswhoseobjectivefunctionis 0 .Thisimpliesthatthe inducedsubgraphisedgeless,i.e.eachofthe K componentsconsistsofasinglenode. Hence,the K remainingnodesformanindependentsetof G ,resultingina`yes'instance fortheK-INDEPENDENTSETPROBLEM.Thustheproofiscomplete. Whenstudyingcombinatorialproblems,integerprogrammin gmodelsareusuallyquite helpfulforprovidingsomeoftheformalpropertiesofthepr oblem[ 138 ].Withthisinmindwe nowdevelopalinearintegerprogrammingformulationforth eCNP. Tobeginwith,denethesurjection u : V V 7!f 0 ; 1 g asabove.Further,weintroducea surjection v : V 7!f 0 ; 1 g denedby v i := 8>><>>: 1 ,ifnode i isdeletedintheoptimalsolution, 0 ,otherwise. (3–1) ThentheCRITICALNODEPROBLEMadmitsthefollowingintegerprogrammingformulation (CNP-1) Minimize X i;j 2 V u ij (3–2) s.t. u ij + v i + v j 1 ; 8 ( i;j ) 2 E; (3–3) u ij + u jk u ki 1 ; 8 ( i;j;k ) 2 V; (3–4) u ij u jk + u ki 1 ; 8 ( i;j;k ) 2 V; (3–5) u ij + u jk + u ki 1 ; 8 ( i;j;k ) 2 V; (3–6) X i 2 V v i k; (3–7) u ij 2f 0 ; 1 g ; 8 i;j 2 V; (3–8) v i 2f 0 ; 1 g ; 8 i 2 V: (3–9) 42


Theorem6. CNP-1 isacorrectformulationfortheCRITICALNODEPROBLEM. Proof. First,wenotethattheobjectiveistondthesetof k nodeswhoseremovalresultsina graphwhichhasthemaximumnumberofdisconnectedcomponen ts.Thisisaccomplishedby theobjectivefunction.Noticethattherstsetofconstrai ntsin( 3–3 )impliesthatifnodes i and j areindifferentcomponentsandifthereisanedgebetweenth em,thenoneofthemmustbe deleted.Furthermore,constraints( 3–4 )-( 3–6 )togetherimplythatforalltripletsofnodes i;j;k; thatif i and j areinsamecomponentand j and k areinsamecomponent,then k and i mustbein thesamecomponent.Constraint( 3–7 )ensuresthatthetotalnumberofdeletednodesislessthan orequalto k .Finally,( 3–8 )and( 3–9 )denetheproperdomainsforthevariablesused.Thus, asolutiontotheintegerprogrammingformulation CNP-1 characterizesafeasiblesolutionto theCNP.Ontheotherhand,itisclearthatafeasiblesolutiontotheCNPwilldeneatleastone feasiblesolutionto CNP-1 .Therefore, CNP-1 isacorrectformulationfortheCNP. Noticethattheconditionswhichsatisfythecircularconst raints( 3–4 ),( 3–5 ),and( 3–6 )in CNP-1 canbesatisedbythesingleconstraint u ij + u jk + u ki 6 =2 ; 8 ( i;j;k ) 2 V: Thuswehave anequivalent,morecompactintegerprogramgivenas (CNP-2) Minimize X i;j 2 V u ij (3–10) s.t. u ij + v i + v j 1 ; 8 ( i;j ) 2 E; (3–11) u ij + u jk + u ki 6 =2 ; 8 ( i;j;k ) 2 V; (3–12) X i 2 V v i k; (3–13) u ij 2f 0 ; 1 g ; 8 i;j 2 V; (3–14) v i 2f 0 ; 1 g ; 8 i 2 V; (3–15) where u i;j and v i areasdenedabove. Noticethatiftheobjectivefunctionhadonlythenumberofc omponents,thenan approximationfortheMAXIMUMK -CUTPROBLEM[ 79 112 ]couldbeemployedbymodifying 43


thecostfunctionoftheGomory-Hutree[ 89 ].Anevensimplerapproachwouldbetoidentify thecutverticesinthegraph,ifanyexist.However,theobje ctivefunctionalsoinvolvesthesizes ofthecomponentsformed,whichmakestheproblemharderand subsequentlyimpliesthatthe methodssuggestedabovearenotsuitableforourproblem. Recallthat P i;j 2 V u ij isameasureofthetotaldisconnectivityofthegraph.Ifweo bserve carefully,theobjectivefunctioncouldberewrittenas X i 2 S s i ( s i 1) 2 ; (3–16) where S issetofallcomponentsand s i isthesizeofthe i thcomponent,whichcanbeeasily identiedbyfastalgorithmslikebreadthordepthrstsear chalgorithmsin O ( j V j + j E j ) time [ 48 ].Wenowprovideanintuitiveexplanationforthechoiceofo urobjectivefunction.Foraxed numberofcomponentsthevarianceinthesizesofthecompone ntswillbethesumofthesquares ofdeviationofsizesofthecomponentsfromthemeansizeofa component.Howevernoticethat themeansizeofanycomponentisconstantbecausethesumoft hesizesofthecomponentsis theconstant, j V j k .Thusminimizingthevarianceofthesizeofthecomponentsr educesto minimizingthesumofsquaresofthesizesofthecomponents, whichisourobjectivefunction. Also,whenthesizesofthecomponentsareequaltheobjectiv efunctionistheminimumwhen thenumberofcomponentsisthemaximum.Wewillusethisobje ctivefunctioninthefollowing sectiontoimplementaheuristicforidentifyingcriticaln odes. 3.2.2CardinalityConstrainedProblem Wenowprovidetheformulationforaslightlymodiedversio noftheCNPbasedon constrainingtheconnectivityindexofthenodesinthegrap h.Givenagraph G =( V;E ) the connectivityindex ofanodeisdenedasthenumberofnodesreachablefromthatv ertex. ExamplesareprovidedinFigure 3-1 .Toconstrainthenetworkconnectivityinoptimization models,wecanimposeconstraintsontheconnectivityindic es. ThisleadstoacardinalityconstrainedversionoftheCNPwhichweaptlyrefertoas theCARDINALITYCONSTRAINEDCRITICALNODEDETECTIONPROBLEM(CC-CNP).The 44


Figure3-1:ConnectivityIndexofnodesA,B,C,Dis3.Connec tivityIndexofE,F,Gis2. ConnectivityIndexofHis0.objectiveistodetectasetofnodes A V suchthattheconnectivityindicesofthenodesin thevertexdeletedsubgraph G ( V n A ) islessthansomethresholdvalue,say L .Usingthesame denitionofthevariablesasintheprevioussubsection,we canformulatetheCC-CNPasthe followingintegerlinearprogrammingproblem. (CC-CNP-1) Minimize X i 2 V v i (3–17) s.t. u ij + v i + v j 1 ; 8 ( i;j ) 2 E; (3–18) u ij + u jk + u ki 6 =2 ; 8 ( i;j;k ) 2 V; (3–19) X i;j 2 V u ij L; (3–20) u ij 2f 0 ; 1 g ; 8 i;j 2 V; (3–21) v i 2f 0 ; 1 g ; 8 i 2 V; (3–22) where L isthemaximumallowableconnectivityindexforanynodein V Theorem7. CC-CNP1 isacorrectformulationfortheCARDINALITYCONSTRAINEDCRITICAL NODEDETECTIONPROBLEM. Proof. ThisprooffollowsinmuchthesamewayasTheorem 6 .First,weseethattheobjective functiongivenclearlyminimizesthenumberofnodesdelete d.Constraints( 3–18 )and( 3–19 ) followexactlyasintheCNPformulation.Theonlydifferenceisnowwemustconstrain 45


procedure CriticalNode ( G;k ) 1 MIS MaximalIndepSet ( G ) 2 while ( j MIS j6 = j V j k ) do 3 i argmin P i 2 S s i ( s i 1) 2 : S 2 G ( MIS [f i g ) ;i 2 V n MIS 4 MIS MIS [f i g 5 endwhile 6 return V n MIS/ setof k nodestodelete / endprocedure CriticalNode Figure3-2:Heuristicfordetectingcriticalnodes. theconnectivityindexofeachnode.Thisisaccomplishedby constraint( 3–20 ).Finally constraints( 3–21 )and( 3–22 )denethedomainsofthedecisionvariables,andwehavethe proof. 3.3HeuristicsforCriticalNodeProblems 3.3.1CNPHeuristic Pseudo-codefortheproposedheuristicisprovidedinFigur e 3-2 .Tobeginwith,the algorithmndsamaximalindependentset(MIS).Theninthel oopfromlines 2 5 ,theheuristic greedilyselectsthenode i 2 V notcurrentlyinMISwhichreturnstheminimumobjective functionforthegraph G ( MIS [f i g ) .Theset MIS isaugmentedtoincludenode i ,andthe processrepeatsuntil j MIS j = j V j k .Themethodterminatesandthesetofcriticalnodestobe deletedisgivenasthosenodes j 2 V suchthat j 2 V n MIS. Theintuitionbehindusinganindependentsetisthatthesub graphinducedbythissetis empty.Statedotherwise,thedeletionofnodesthatare not intheindependentsetfromthegraph willresultinanemptysubgraph.Noticethatthiswillprovi detheoptimalsolutionforaninstance oftheCNPif j MIS jj V j k .However,ifthesizeofMISislessthan j V j k ,wesimply keepaddingnodeswhichprovidethebestobjectivevaluetot hesetuntilitreachesthedesired size.Inthefollowinglemma,weestablisharelationshipbe tweentheCNPandtheMAXIMUM INDEPENDENTSETproblem,whichalsoprovidesaboundontheoptimalsolution foraninstance oftheCNP. 46


Lemma5. Givenagraph G =( V;E ) ,thecardinalityofthemaximumindependentsetof G denoted ( G ) providesanupperboundonthenumberofcomponentsproduced intheoptimal solutionofthecorrespondingCRITICALNODEPROBLEMforanyvalueof k 2 Z Proof. Obviously,removingthecriticalnodesdeterminedbytheop timalsolutionforany instanceoftheCNPresultsinasetofdisconnectedcomponentsof G .Onenodefromeach ofthesecomponentsformsanindependentset.Hence ( G ) shouldbeatleastaslargeasthe numberofcomponentsformedintheoptimalsolutiontotheCNP.Furthermore,thecomponents formedinthesubgraphinducedbythemaximumindependentse tareofsizeone,andhence resultintheoptimalsolutionfortheCNPinstanceif ( G ) j V j k ,i.e.ifthedeletionofsome k nodesresultsinanemptygraph.Thus,wehavethelemma. Wenotethatthisboundisnotparticularlyusefulinpractic esincetheMAXIMUM INDEPENDENTSETproblemis NP -hardingeneral[ 24 79 ].However,amaximalindependent setcanbecomputedinpolynomialtime.Thismotivatesourde cisiontouse maximal instead of maximum independentsetsintheheuristic.Subsequentlytheheuris ticiscomputationally efcient,withthecomplexitygiveninthefollowingtheore m. Theorem8. Theproposedalgorithmhascomplexity O ( k 2 + j V j k )) Proof. Tobeginwith,the while loopfromlines 2 5 williterateatmost O ( j V j k ) times.Ineach iteration,thenumberofsearchoperationsdecreasesfrom j V j 1 to j V j ( j V j k )= k .Note thatweareperformingthesearchofasparsegraph,whichisi nitiallyempty.Hencethetotal complexitywillbe O ( j V j 1+ j V j 2+ + j V jj V j + k )= O 0@ j V j X i =1 i j V j k X i =1 i 1A = O ( k 2 + j V j k ) : Thustheproofiscomplete. TheproposedalgorithmndsafeasiblesolutiontotheCRITICALNODEPROBLEM; however,thesolutionisnotguaranteedtobegloballyorloc allyoptimal.Therefore,wecan enhancetheheuristicwiththeapplicationoflocalsearchr outineasfollows.Considerthe 47


procedure LocalSearch ( V n MIS) 1 X MIS 2 local improvement : TRUE : 3 while local improvement do 4 local improvement : FALSE : 5 if i 2 MIS and j 62 MIS then 6 MIS MIS n i 7 MIS MIS [ j 8 if f (MIS)

procedure CriticalNodeLS ( G;k ) 1 X ; 2 f ( X ) 1 3 for j =1 to MaxIter do 4 X CriticalNode ( G;k ) 5 X LocalSearch (X) 6 if f ( X )

procedure ConstrainedCriticalNode ( G;L ) 1 MIS MaximalIndepSet ( G ) 2 OPT FALSE 3 NoAdd 0 4 while (OPT : NOT : TRUE ) do 5 for ( i =1 to j V j ) do 6 if j s j ( j s j 1) 2 L 8 s 2 S G ( MIS [f i g ): i 2 V n MIS then 7 MIS MIS [f i g 8 else 9 NoAdd NoAdd +1 10 endif 11 if ( NoAdd = j V jj MIS j ) then 12 OPT TRUE 13 BREAK 14 endif 15 endfor 16 endwhile 17 return V n MIS/ setofnodestodelete / endprocedure ConstrainedCriticalNode Figure3-5:HeuristicfortheCARDINALITYCONSTRAINEDCRITICALNODEPROBLEM. 7 ,otherwiseNoAddisincremented.IfNoAddiseverequalto j V jj MIS j ,thennonodescanbe returnedtothegraphandOPTissetto TRUE .Thenloopisthenexitedandthealgorithmreturns thesetofnodestobedeleted,i.e. V n MIS. Theorem11. Theworst-casecomplexityofthe ConstrainedCriticalNode heuristicis O ( j V j 2 + j V jj E j ) Proof. ThisproofissimilartotheproofofTheorem 8 above.Theloopfromlines 4 16 will iterateatmost O ( j V j ) times.Eachlooprequiresatmost O ( j V j + j E j ) timetoverifythe ifasolutionwillremainfeasibleafteranodeisre-include dinthegraph.Thuswehavethe result. 3.3.3GeneticAlgorithmfortheCC-CNPAsmentionedinSubsection 2.7.2 ,geneticalgorithms(GAs)mimicthebiologicalprocessof evolution.Inthissubsection,wedescribetheimplementat ionofaGAfortheCC-CNP.Recallthe generalstructureofaGAasoutlinedinFigure 3-6 .Whendesigningageneticalgorithmforan 50


procedure GeneticAlgorithm 1 Generatepopulation P k 2 Evaluatepopulation P k 3 while terminatingconditionnotmet do 4 Selectindividualsfrom P k andcopyto P k +1 5 Crossoverindividualsfrom P k andputin P k +1 6 Mutateindividualsfrom P k andputin P k +1 7 Evaluatepopulation P k +1 8 P k P k +1 9 P k +1 ; 10 endwhile 11 return bestindividualin P k endprocedure GeneticAlgorithm Figure3-6:Pseudo-codeforagenericgeneticalgorithm. optimizationproblem,onemustprovideameanstoencodethe population,denethecrossover operator,anddenethemutationoperatorwhichallowsforr andomchangesinoffspringtohelp preventthealgorithmfromconvergingprematurely[ 10 ]. Forourimplementation,weusebinaryvectorsasanencoding schemeforindividualswithin thepopulationofsolutions.Whenthepopulationisgenerat ed,(Figure 3-6 ,line 1 ),arandom deviatefromadistributionwhichisuniformonto (0 ; 1) 2 R isgeneratedforeachnode.Ifthe deviateexceedssomespeciedvalue,thecorrespondingall eleisassignedvalue 1 ,indicating thisnodeshouldbedeleted.Otherwise,thealleleisgivena 0 ,implyingitisnotdeleted.Inorder toevaluatethetnessofthepopulation,perline 2 ,wemustdeterminewhethereachindividual solutionisfeasibleornot.Determiningfeasibilityisare lativelystraightforwardtaskandcan accomplishedin O ( j V j + j E j ) usingadepth-rstsearch[ 6 ]. Inordertoevolvethepopulationoversuccessivegeneratio ns,weuseareproductionscheme inwhichtheparentschosentoproducetheoffspringaresele ctedusingthebinarytournament method[ 131 172 ].Usingthismethod,twochromosomesarechosenatrandomfr omthe populationandtheonehavingthebesttness,i.e.thelowes tobjectivefunctionvalue,iskept asaparent.Theprocessisthenrepeatedtoselectthesecond parent.Thetwoparentsarethen combinedusingacrossoveroperatortoproduceanoffspring [ 94 ]. 51


CoinToss T H H T H MOM 0 : 56 0 : 81 0 : 22 0 : 7 0 : 86 DAD 0 : 29 0 : 49 0 : 98 0 : 12 0 : 32 Offspring 0 : 29 0 : 81 0 : 22 0 : 12 0 : 86 Figure3-7:Exampleofthecrossoveroperation.Inthiscase CrossProb =0 : 65 Tobreednewsolutions,weimplementastrategyknownas parameterizeduniformcrossover [ 167 ].Thismethodworksasfollows.Aftertheselectionofthepa rents,refertotheparent havingthebesttnessas MOM .Foreachofthenodes(alleles),abiasedcoinistossed.Ift he resultisheads,thentheallelefromthe MOM chromosomeischosen.Otherwise,theallelefrom theleasttparent,callit DAD ,isselected.Theprobabilitythatthecoinlandsonheadsis known as CrossProb ,andisdeterminedempirically.Figure 3-7 providesanexampleofapotential crossoverwhenthenumberofnodesis 5 and CrossProb =0 : 65 [ 10 ]. Afterthechildisproduced,themutationoperatorisapplie d.Mutationisarandomizing agentwhichhelpspreventtheGAfromconvergingprematurel yandescapetolocaloptima. Thisprocessworksbyippingabiasedcoinforeachalleleof thechromosome.Theprobability ofthecoinlandingheads,knownasthemutationrate( MutRate )istypicallyaverysmalluser denedvalue.Iftheresultisheads,thenthevalueofthecor respondingalleleisreversed.Forour implementation, MutRate =0 : 03 Afterthecrossoverandmutationoperatorscreatethenewof fspring,itreplacesacurrent memberofthepopulationusingtheso-called steady-state model[ 37 94 131 ].Usingthis methodology,thechildreplacestheleasttmemberofthepo pulation,providedthataclone ofthechildisnotanexistingmemberinthepopulation.This methodensuresthattheworst elementofthepopulationismonotonicallyimprovingineve rygeneration.Inthesubsequent iteration,thechildbecomeseligibletobeaparentandthep rocessrepeats.ThoughtheGAdoes convergeinprobabilitytotheoptimalsolution,itiscommo ntostoptheprocedureaftersome “terminatingcondition”(Figure 3-6 ,line 3 )issatised.Thisconditioncouldbeoneofseveral thingsincluding,amaximumrunningtime,atargetobjectiv evalue,oralimitonthenumberof 52


Table3-1:ResultsofIPmodelandheuristiconterroristnet workdata. Instance IPModel Heuristic Heuristic+LS Nodes ObjectiveExecution ObjectiveExecution ObjectiveExecution Deleted( k ) ValueTime(s) ValueTime(s) ValueTime(s) 20 2012 : 69 220 : 08 200 : 01 15 61277 : 77 660 : 03 610 : 01 10 1693337 : 06 1900 : 06 1690 : 02 9 2142792 : 33 2290 : 15 2140 : 02 8 28215111 : 94 3090 : 04 2820 : 01 7 32710792 : 08 3290 : 09 3270 : 01 generations.Forourimplementation,weusethelatteropti onandthebestsolutionafter MaxGen generationsisreturned. 3.4ComputationalResults Alloftheproposedheuristicswereimplementedinthe C ++programminglanguageand compliedusingGNU g ++version 3 : 4 : 4 ,usingoptimizationagsO2 .Itwastestedona PC equippedwitha1700MHzIntel R r Pentium R r Mprocessorand 1 : 0 gigabytesofRAMoperating undertheMicrosoft R r Windows R r XPProfessionalenvironment. 3.4.1CNPResults Webeginwiththenumericalresultsofthecombinatorialalg orithmfortheCRITICALNODE PROBLEM.WetestedtheIPmodelandtheaforementionedheuristicont heterroristnetwork fromKrebs[ 118 ]aswellasonasetofrandomlygeneratedscale-free[ 13 ]graphsranginginsize from 75 to 150 nodeswithvariousdensities.Thegraphsweregeneratedwit hversion 1 : 4 ofthe publiclyavailableBarabasigraphgeneratorbyDreier[ 62 ].Foreachinstancetested,wereport solutionsfor 3 valuesof k ,thenumberofnodestobedeleted. Asabasisforcomparison,wehaveimplementedtheintegerpr ogrammingmodelfortheCRITICALNODEPROBLEMusingtheCPLEX TM optimizationsuitefromILOG[ 50 ].CPLEX containsanimplementationofthesimplexmethod[ 98 ],andusesabranchandboundalgorithm [ 173 ]togetherwithadvancedcutting-planetechniques[ 107 139 ]. Webeginbyprovidingtheresultsfromtheterroristnetwork [ 118 ].Thegraph,whichis showninFigure 3-8 has 62 nodesand 153 edges.Noticethatnode 38 isthecentralnodewith 53


Figure3-8:TerroristnetworkcompiledbyKrebs. degree 22 .WeappliedtheIPformulationandtheheuristictothisnetw orkwith 6 valuesof k TheresultsareprovidedinTable 3-1 .Noticethatforallvaluesof k ,theheuristiccomputedthe optimalsolutionrequiringonaverage 0 : 013 secondsofcomputationtime.Theaveragetimeto computetheoptimalsolutionusingCPLEXwas 5387 : 31 seconds.Clearlyevenforthisrelatively smallnetwork,theheuristicisthemethodofchoice.Figure 3-9 showstheresultinggraphofthe terroristnetworkaccordingtotheoptimalsolutiontotheCNPfortheinstanceof k =20 Inordertodeterminethescalabilityandrobustness,thepr oposedheuristicwastestedona setofrandomlygeneratedscale-freegraphs.Table 3-2 presentstheresultsoftheheuristicandthe optimalsolverwhenappliedtotherandominstances.Foreac hinstance,wereportthenumber ofnodesandarcs,thevalueof k beingconsidered,theoptimalsolutionandcomputationtim e requiredbyCPLEX,andnallytheheuristicsolutionandthe correspondingcomputationtime. Foreachgraph,wereportsolutionsfor 3 differentvaluesof k 54


Figure3-9:Optimalsolutionwhen k =20 Noticethatforallinstancestested,ourmethodwasabletoc omputetheoptimalsolution. Furthermore,therequiredtimetocomputetheoptimalsolut ionwaslessthanonesecondforall butoneinstance,averagingonly 0 : 33 secondsforall 27 instances.Ontheotherhand,CPLEX required 289 : 44 secondsonaveragetocomputetheoptimalsolution,requiri ngover 5000 seconds intheworstcase.Ourcomputationalexperimentsindicatet hattheproposedheuristicisableto efcientlyprovideexcellentsolutionsforlarge-scalein stancesoftheCNP. 3.4.2CC-CNPResults Wecontinuewiththeresultsofthetwoalgorithmsdeveloped fortheCC-CNP,namely thecombinatorialalgorithmandthegeneticalgorithm.Asa bove,wetestedtheIPmodeland bothheuristicsontheterroristnetwork[ 118 ]andasetofrandomlygeneratedgraphs.Foreach instancetested,wereportsolutionsfor 3 valuesof L ,theconnectivityindexthreshold.Finally, wehaveimplementedtheintegerprogrammingmodelfortheCC-CNPusingCPLEX TM 55


Table3-2:ResultsofIPmodelandheuristiconrandomlygene ratedscalefreegraphs. Instance IPModel Heuristic Heuristic+LS NodesArcsDeleted ObjComp ObjComp ObjComp Nodes( k ) ValueTime(s) ValueTime(s) ValueTime(s) 7514020 3666 : 7 920 : 12 360 : 03 7514025 1833 : 28 390 : 28 180 : 03 7514030 74 : 23 180 : 02 70 : 04 7521025 2693 : 71 780 : 1 260 : 04 7521030 83 : 57 310 : 05 80 : 05 7521035 24 : 36 160 : 18 20 : 04 7528033 26749 : 19 540 : 00 260 : 04 7528035 20164 : 34 380 : 09 200 : 06 7528037 1383 : 98 240 : 39 130 : 11 10019425 44151 : 14 1420 : 731 440 : 09 10019430 2059 : 66 720 : 56 200 : 11 10019435 108 : 51 330 : 66 100 : 12 10028540 23136 : 47 481 : 151 230 : 11 10028542 17263 : 82 380 : 4 170 : 17 10028545 1116 : 78 290 : 53 110 : 23 10038045 22128 : 13 580 : 58 220 : 15 10038047 16243 : 07 421 : 191 160 : 16 10038050 10228 : 72 230 : 31 100 : 11 12524033 625047 : 51 970 : 721 620 : 30 12524040 29118 : 92 491 : 562 290 : 24 12524045 1617 : 09 320 : 14 160 : 39 15029040 4041 : 6 1251 : 832 400 : 47 15029050 1226 : 29 642 : 773 120 : 831 15029060 124 : 92 351 : 091 10 : 851 15043561 1929 : 55 532 : 313 190 : 741 15043565 1331 : 45 370 : 991 131 : 952 15043567 1137 : 91 310 : 52 110 : 801 Table 3-3 presentscomputationalresultsoftheIPmodelandheuristi csolutionswhentested ontheterroristnetworkdata.Noticethatforall 5 valuesof L tested,thegeneticalgorithmand thecombinatorialalgorithmwithlocalsearch(ComAlg+LS) computedoptimalsolutions. Figure 3-10 showstheoptimalsolutionforthecasewhen L =4 Wenowconsidertheperformanceofthealgorithmswhenteste dontherandomlygenerated datasetscontainingupto 50 nodestakenfrom[ 9 ].TheresultsareshowninTable 3-4 .Forthese relativelysmallinstances,wewereabletocomputetheopti malsolutionsusingCPLEX.Foreach 56


Table3-3:ResultsofIPmodelandheuristicsonterroristne tworkdata. Instance IPModel GeneticAlg ComAlg ComAlg+LS MaxConn. ObjComp ObjComp ObjComp ObjComp Index( L ) ValTime(s) ValTime(s) ValTime(s) ValTime(s) 3 21188 : 98 210 : 25 220 : 01 210 : 1 4 17886 : 09 170 : 741 190 : 01 170 : 45 5 1530051 : 09 150 : 871 200 : 18 251 : 331 8 130 : 39 140 : 05 130 : 07 10 110 : 741 120 : 07 110 : 05 Figure3-10:Optimalsolutionwhen L =4 instance,weprovidesolutionsfor 3 valuesof L ,themaximumconnectivityindex.Noticethatfor theseproblems,thegeneticalgorithmcomputedoptimalsol utionsforeachinstancetestedina fractionofthetimerequiredbyCPLEX.Thecombinatorialhe uristicfoundoptimalsolutionsfor allbut 3 casesrequiringapproximatelyhalfofthetimeoftheGA. Table 3-5 presentsthesolutionsfortherandominstancesfrom 75 to 150 nodes[ 9 11 ]. Again,inordertodemonstratetherobustnessoftheheurist ics,weprovidesolutionsfor 3 57


Table3-4:ResultsoftheIPmodelandgeneticalgorithmandt hecombinatorialheuristicon randomlygeneratedscalefreegraphs. Instance IPModel GeneticAlg ComAlg+LS NodesArcsMaxConn. ObjComp ObjComp ObjComp Index( L ) ValueTime(s) ValueTime(s) ValueTime(s) 20452 90 : 04 90 : 02 90 : 03 20454 60 : 13 60 : 04 60 : 862 20458 50 : 39 50 : 04 51 : 482 25602 110 : 07 110 : 49 110 : 08 25604 914 : 1 92 : 113 100 : 01 25608 726 : 64 70 : 05 80 : 06 30502 110 : 07 110 : 06 110 : 01 30504 80 : 1 80 : 05 80 30508 61152 : 15 60 : 09 60 30754 1018 : 77 100 : 14 100 : 02 30756 9442 : 41 90 : 09 90 : 04 307510 764 : 94 70 : 18 80 35602 120 : 13 120 : 14 120 : 14 35604 829 : 89 80 : 711 80 35606 731 : 61 70 : 31 70 : 01 40702 150 : 17 150 : 1 150 : 101 40704 11341 : 97 110 : 06 110 40706 878 : 94 80 : 2 80 : 04 45802 160 : 24 160 : 06 160 : 1 45804 1148 : 17 110 : 05 110 : 02 45806 8118 : 23 80 : 09 80 : 071 501352 190 : 36 190 : 27 190 : 05 501354 15165 : 18 150 : 63 150 : 291 501356 145722 : 88 140 : 721 140 : 03 Total(Sum) 248257 : 58 246 : 705 273 : 417 valuesof L foreachinstance.Inthistable,weprovidetheresultsfort hegeneticalgorithmand combinatorialheuristicwithandwithoutthelocalsearche nhancement.CPLEXwasunableto computeoptimalsolutionswithinreasonabletimelimitsfo ranyoftheinstancesrepresentedin thistable. Weseefromthistablethattheintermsofsolutionqualityth eGAisthebestperforming method.The ComAlg + LS alsofavorswell,butrequiresmorecomputationtimethanth eGAand requiresmorecomputingtimeonaverage.Thecombinatorial algorithmwithoutthelocalsearch procedureproducessolutionwhicharearguablyreasonable giventhattherequiredcomputation timeisover 36 timesfasterthantheGA,whilethesolutionsareonly 1 : 2 timesworsethanthose 58


Table3-5:Comparativeresultsofthegeneticalgorithmand thecombinatorialheuristicwhen testedonthelargerrandomgraphs.Duetothecomplexity,we wereunabletocomputethe correspondingoptimalsolutions. Instance GeneticAlgorithm ComAlg ComAlg+LS NodesArcsMaxConn. ObjComp ObjComp ObjComp Index( L ) ValueTime(s) ValueTime(s) ValueTime(s) 751405 181 : 622 210 181 : 502 751408 141 : 442 200 : 02 141 : 181 7514010 121 : 231 200 : 12 123 : 364 752105 231 : 532 290 : 01 2318 : 476 752108 212 : 443 230 : 01 222 : 934 7521010 202 : 794 240 : 09 2021 : 17 752805 313 : 464 350 : 101 313 : 144 752808 292 : 874 310 : 05 293 : 746 7528010 283 : 775 300 : 13 284 : 787 1001945 225 : 317 330 : 02 222 : 774 10019410 173 : 224 220 : 241 176 : 499 10019415 152 : 954 220 : 021 150 : 44 1002855 335 : 08 380 : 02 331 : 262 10028510 284 : 376 310 : 05 2811 : 076 10028515 275 : 728 280 : 16 271 : 142 1003805 409 : 052 470 : 051 425 : 739 10038010 3611 : 506 410 : 02 373 : 866 10038015 356 : 198 400 : 39 363 : 034 1252405 297 : 951 370 : 251 311 : 472 12524010 249 : 984 290 : 07 241 : 993 12524015 225 : 888 260 : 18 229 : 233 1502905 317 : 981 400 : 421 305 : 798 15029010 264 : 967 320 : 2 255 : 107 15029015 235 : 457 291 : 101 2319 : 889 1504355 499 : 143 570 : 06 496 : 459 15043510 4019 : 407 500 : 44 415 : 518 15043515 389 : 703 450 : 07 3813 : 699 Total(Sum) 731155 : 183 8804 : 297 737165 : 304 computedbytheGA.Nevertheless,thegeneticalgorithmreq uiredonly 5 : 748 secondsonaverage tocomputethebestsolution.Thetrade-offofsolutionqual ityversuscomputationtimeisa decisionthatwouldbemadebyanoperatordependingonthesi zeofthenetworkandthetime constraintsimposedondetectingthecriticalnodesofagiv engraph. 3.5ConcludingRemarks Inthischapter,weproposedseveralmethodsofjammingcomm unicationnetworksbased onthedetectionofthecriticalnodes.Criticalnodesareth oseverticeswhosedeletionresultsin 59


themaximumnetworkdisconnectivity.Ingeneral,theprobl emofdetectingcriticalnodeshas awidevarietyofapplicationsfromjammingcommunicationn etworksandotheranti-terrorism applications,toepidemiologyandtransportationscience [ 9 11 ]. Inparticularweexaminedtwoproblems,namelytheCRITICALNODEPROBLEM(CNP)as wellastheCARDINALITYCONSTRAINEDCNP(CC-CNP).Givenagraphandaninteger k ,the objectiveoftheCNPistodetectasetof k criticalnodeswhosedeletionresultsinthemaximum numberofdisconnectedcomponentswhosecardinalitieshav etheminimumvariance.The denitionoftheCC-CNPisslightlydifferentinthatinsteadofgiven k 2 Z ,themaximumnumber ofnodestodelete,wearegivensomevalue L 2 Z whichrepresentsthemaximumconnectivity indexanodemayhave.Theobjectiveinthiscaseistodeletet heminimumnumberofnodes whileensuringthattheconnectivityindexofeachnodedoes notexceed L Theproposedproblemsweremodeledasintegerlinearprogra mmingproblems.Thenwe provedthatthecorrespondingdecisionproblemsare NP -complete.Furthermore,weproposed aseveralheuristicsforefcientlycomputingqualitysolu tionstolarge-scaleinstances.The heuristicproposedfortheCNPwasacombinatorialalgorithmwhichexploitedpropertieso f thegraphinordertocomputebasicfeasiblesolutions.Them ethodwasfurtherintensiedby theapplicationofalocalsearchmechanism.Byusingtheint egerprogrammingformulation wewereabletodeterminetheprecisionofourheuristicbyco mparingtheirrelativesolutions andcomputationtimesforseveralnetworks.Thecomputatio nalexperimentsindicatedthatthe heuristicfoundoptimalsolutionsforallinstancestested inafractionofthetimerequiredbythe commercialIPsolverCPLEX. FortheCC-CNPweproposedtwoalgorithms,namelyamodiedversionofthe combinatorialalgorithmdescribedaboveandageneticalgo rithm[ 87 ].Onceagain,the computationalexperimentsindicatedthatbothmethodsare robustandareabletoefciently computeapproximatesolutionsforinstancesupto 150 nodes. Wealsoconcludewithafewwordsonthepossibilityoffuture expansionofthiswork.A heuristicexplorationofcuttingplanealgorithmsontheIP formulationwouldbeaninteresting 60


alternative.Otherheuristicapproachesworthyofinvesti gationincludehybridizingthegenetic algorithmwiththeadditionofalocalsearchorpath-relink ingenhancementprocedure[ 85 ]. Finally,thelocalsearchusedinthecombinatorialalgorit hmwasasimple 2 -exchangemethod, whichwasthecauseofasignicantslowdownincomputationa snotedinTable 3-5 .Amore sophisticatedlocalsearchsuchasamodicationoftheonep roposedbyResendeandWerneck [ 159 160 ]shouldbeamajorfocusofattention. Furthermore,itwouldbeinterestingtostudytheweightedv ersionoftheproblemtosee howweightsaddedtothenodesaffectthesolutions.Forexam ple,itisrationaltoperceive applicationscontainingweightednetworksinwhichthecos tofdeletingonenodeisdifferent fromanother.Also,pertainingtoapplicationsoutsidethe scopeofjammingnetworks,astudyof epidemicthresholdvariationwithrespecttotheheuristic resultswillhelpdeterminetheimpacts oncontagionsuppressioninbiologicalandsocialnetworks 61


CHAPTER4 THEWIRELESSNETWORKJAMMINGPROBLEM 4.1Introduction Militarystrategistsareconstantlyseekingwaystoincrea setheeffectivenessoftheir forcewhilereducingtheriskofcasualties.Inanyadversar ialenvironment,animportant goalisalwaystoneutralizethecommunicationsystemofthe enemy.Inthischapter,weare interestedinjammingawirelesscommunicationnetwork.Sp ecically,westudytheproblem ofdeterminingtheoptimalnumberandplacementforasetofj ammingdevicesinorderto neutralizecommunicationonthenetwork.Thisisknownasth eWIRELESSNETWORKJAMMING PROBLEM(WNJP). Despitetheenormousamountofresearchontelecommunicati onsystems[ 155 ],thetopicof jammingcommunicationnetworkshasreceivedlittleattent ion.Infact,thematerialthatfollows inthenexttwochapterspresenttherstsuchefforts,insof araswecantell.Wewillbegin thischapterbydescribingandformulatingtheproblemofja mmingawiredtelecommunication network,andextendthisresulttothewirelessdomain.Wewi llseethatthereisabitmore versatilitywhenconsideringthewirelessversionofthepr oblemduetothewirelessmulticast advantage,i.e.theabilityofwirelesstransmitterstocom municateaffectnodesthatarenot directlyadjacenttothem. Wecangeneralizetheworkof[ 9 ]tostudytheproblemofjammingandeavesdropping wirelesscommunicationnetworks.Aswewillsee,thereares everalvariationsthatcanbemade dependingontheoverallobjectives.Thisisaidedbythefac tthatwirelessjammingdevices notonlyaffectthosenodeswhicharedirectlyadjacenttoth em;rather,theypropagateenergy throughoutthenetworktoallthecommunicationnodesaswew illseeinthenextsection. Theorganizationofthechapterisasfollows.Afterareview ofrelatedwork,wepresent severaldeterministicformulationsoftheWNJPinSection 4.3 .Inparticular,Subsection 4.3.1 containsseveralcoverageformulationsoftheWNJP.TheninSubsection 4.3.2 ,weuse toolsfromgraphtheorytodenetheconnectivityofthenetw orkanddevelopanalternative 62


formulationbasedonconstrainingtheconnectivityindice softhenodes,analogoustotheCC-CNP.Next,inSection 4.4 weincorporatepercentileconstraintstodevelopformulat ions whicharecomputationallymoreefcientandhavesimilarso lutionquality.InSection 4.5 ,we willpresenttwocasestudiescomparingthesolutionsandco mputationtimeforallformulations. Finally,conclusionsandfuturedirectionsofresearchwil lbeaddressed. 4.2DenitionsandAssumptions Beforeformallydeningtheproblemstatement,wewillstat esomebasicassumptionsabout thejammingdevicesandthecommunicationnodesbeingjamme d.Weassumethatparameters suchasthefrequencyrangeofthejammingdevicesareknown. Inaddition,thejammingdevices areassumedtohaveomnidirectionalantennas.Thecommunic ationnodesarealsoassumedtobe outttedwithomnidirectionalantennasandfunctionasbot hreceiversandtransmitters.Givena graph G =( V;E ) ,wecanrepresentthecommunicationdevicesasthevertices ofthegraph.An undirectededgewouldconnecttwonodesiftheyarewithinac ertaincommunicationthreshold. Givenaset M = f 1 ; 2 ;:::;m g ofcommunicationnodestobejammed,thegoalisto ndasetoflocationsforplacingjammingdevicesinorderto suppressthefunctionalityofthe network.The jammingeffectiveness ofdevice j iscalculatedusing d :( V V ) 7! R ,where d isadecreasingfunctionofthedistancefromthejammingdev icetothenodebeingjammed. Hereweareconsideringradiotransmittingnodes,andcorre spondingly,jammingdeviceswhich emitelectromagneticwaves.Thusthejammingeffectivenes sofadevicedependsonthepower ofitselectromagneticemission,whichisassumedtobeinve rselyproportionaltothesquared distancefromthejammingdevicetothenodebeingjammed.We notethatthisassumptionis madewithoutthelossofgenerality.Theresultspresentedi nthischapterholdaslongasthe function d isasmoothmonotonicallydecreasingfunction.Specicall y, d ij := r 2 ( i;j ) ; where 2 R isaconstant,and r ( i;j ) representsthedistancebetweennode i andjammingdevice j .Withoutthelossofgenerality,wecanset =1 63


4.3DeterministicFormulations 4.3.1CoverageApproach Thecumulativelevelofjammingenergyreceivedatnode i isdenedas Q i := n X j =1 d ij = n X j =1 1 r 2 ( i;j ) ; where n isthenumberofjammingdevices.Then,wecanformulatetheWIRELESSNETWORK JAMMINGPROBLEM(WNJP)astheminimizationofthenumberofjammingdevicesplaced subjecttoasetof qualitycovering constraints: (QCP) Minimize n (4–1) s.t. Q i C i ;i =1 ; 2 ;:::;m: (4–2) Thesolutiontothisproblemprovidestheoptimalnumberofj ammingdevicesneededto ensureacertainjammingthreshold C i ismetateverynode i 2M .Acontinuousoptimization approachwhereoneisseekingtheoptimalplacementcoordin ates ( x j ;y j ) ;j =1 ; 2 ;:::;n forjammingdevicesgiventhecoordinates ( X i ;Y i ) ;i =1 ; 2 ;:::;m ,ofnetworknodes,leads tohighlynon-convexformulations.Forexample,considert hequalitycoveringconstraintfor networknode i n X j =1 1 ( x j X i ) 2 +( y j Y i ) 2 C i : Itiseasytoverifythatthisconstraintisnon-convex.Find ingtheoptimalsolutiontothis nonlinearprogrammingproblemwouldrequireanextensivea mountofcomputationaleffort. Toovercomethenon-convexityoftheaboveformulation,wep roposeseveralinteger programmingmodelsfortheproblem.Supposenowthatalongw iththesetofcommunication nodes M = f 1 ; 2 ;:::;m g ,thereisaxedset N = f 1 ; 2 ;:::;n g ofpossiblelocationsforthe jammingdevices.Thisassumptionisreasonablebecauseinr ealbattleeldscenarios,thesetof 64


possibleplacementlocationswilllikelybelimited.Dene thedecisionvariable x j as x j := 8>><>>: 1 ; ifajammingdeviceisinstalledatlocation j; 0 ; otherwise : (4–3) Ifweredene r ( i;j ) tobethedistancebetweencommunicationnode i andjamminglocation j thenwehavetheOPTIMALNETWORKCOVERING(ONC)formulationoftheWNJPas (ONC) Minimize n X j =1 c j x j (4–4) s.t. n X j =1 d ij x j C i ;i =1 ; 2 ;:::;m (4–5) x j 2f 0 ; 1 g ;j =1 ; 2 ;:::;n; (4–6) where C i isdenedasabove.Heretheobjectiveistominimizethenumb erofjammingdevices usedwhileachievingsomeminimumlevelofcoverageateachn ode.Thecoefcients c j in( 4–4 ) representthecostsofinstallingajammingdeviceatlocati on j .Inabattleeldscenario,placing ajammingdeviceinthedirectproximityofanetworknodemay betheoreticallypossible; however,suchaplacementmightbeundesirableduetosecuri tyconsiderations.Inthiscase,the locationconsideredwouldhaveahigherplacementcostthan wouldasaferlocation.Ifthereare nopreferencesfordevicelocations,thenwithoutthelosso fgenerality, c j =1 ;j =1 ; 2 ;:::;n: Thoughwehaveremovedthenon-convexcoveringconstraints ,thisformulationremains computationallydifcult.NoticethatONCisformulatedasaMULTIDIMENSIONALKNAPSACK PROBLEMwhichisknowntobe NP -hardingeneral[ 79 ]. 4.3.2ConnectivityFormulation InthegeneralWNJP,itisimportantthatthedistinctionbemadethattheobject iveisnot simplytojamallofthenodes,buttodestroythefunctionali tyoftheunderlyingcommunication network.Inthissection,weusetoolsfromgraphtheorytode velopamethodforsuppressing 65


Figure4-1:ConnectivityIndexofnodesA,B,C,Dis3.Connec tivityIndexofE,F,Gis2. ConnectivityIndexofHis0.thenetworkbyjammingthosenodeswithseveralcommunicati onlinksandderiveanalternative formulationoftheWNJP.Givenagraph G =( V;E ) ,therecallthatthe connectivityindex of anodeisdenedasthenumberofnodesreachablefromthatver tex(asshowninFigure 4-1 ). Toconstrainthenetworkconnectivityinoptimizationmode ls,wecanimposeconstraintsonthe connectivityindicesinsteadofusingcoveringconstraint s. WecannowdevelopaformulationfortheWNJPbasedontheconnectivityindicesofthe communicationgraph.Weassumethatthesetofcommunicatio nnodes M = f 1 ; 2 ;:::;m g to bejammedisknownandasetofpossiblelocations N = f 1 ; 2 ;:::;n g forthejammingdevices isgiven.Notethaninthecommunicationgraph, V M .Let S i := P nj =1 d ij x j denotethe cumulativelevelofjammingatnode i .Thennode i issaidtobejammedif S i exceedssome thresholdvalue C i .Wesaythatcommunicationisseveredbetweennodes i and j ifatleastone ofthenodesisjammed.Further,let y : MM7!f 0 ; 1 g beasurjectionwhere y ij :=1 if thereexistsapathfromnode i tonode j inthejammednetwork.Lastly,let z : M7!f 0 ; 1 g bea surjectivefunctionwhere z i returns1ifnode i isnotjammed. TheobjectiveoftheCONNECTIVITYINDEXPROBLEM(CIP)formulationoftheWNJPisto minimizetotaljammingcostsubjecttoaconstraintthatthe connectivityindexofeachnodedoes 66


notexceedsomepre-describedlevel L .Thecorrespondingoptimizationproblemisgivenas: (CIP) Minimize n X j =1 c j x j (4–7) s.t. m X j =1 j 6 = i y ij L; 8 i 2M ; (4–8) M (1 z i ) >S i C i Mz i ; 8 i 2M ; (4–9) x j 2f 0 ; 1 g ; 8 j 2N ; (4–10) z i 2f 0 ; 1 g8 i 2M ; (4–11) 8 i;j 2M ;y ij 2f 0 ; 1 g ; 8 i;j 2M ; (4–12) where M 2 R issomelargeconstant. Let v : MM7!f 0 ; 1 g and v 0 : MM7!f 0 ; 1 g bedenedasfollows: v ij := 8>><>>: 1 ; if ( i;j ) 2 E; 0 ; otherwise, (4–13) and v 0 ij := 8>><>>: 1 ; if ( i;j ) existsinthejammednetwork ; 0 ; otherwise : (4–14) 67


Withthis,wecanformulateanequivalentintegerprogramas (CIP-1) Minimize n X j =1 c j x j ; (4–15) s.t. y ij v 0 ij ; 8 i;j 2M ; (4–16) y ij y ik y kj ;k 6 = i;j ; 8 i;j 2M ; (4–17) v 0 ij v ij z j z i ;i 6 = j ; 8 i;j 2M ; (4–18) m X j =1 j 6 = i y ij L; 8 i 2M ; (4–19) M (1 z i ) >S i C i Mz i ; 8 i 2M ; (4–20) z i 2f 0 ; 1 g ; 8 i 2M ; (4–21) x j 2f 0 ; 1 g ; 8 j 2N ;y ij 2f 0 ; 1 g8 i;j 2M ; (4–22) v ij 2f 0 ; 1 g ; 8 i;j 2M ;v 0 ij 2f 0 ; 1 g ; 8 i;j 2M : (4–23) Lemma6. IfCIPhasanoptimalsolutionthen,CIP-1 hasanoptimalsolution.Further,any optimalsolution x oftheoptimizationproblemCIP-1 isanoptimalsolutionofCIP. Proof. Itiseasytoestablishthatif i and j arereachablefromeachotherinthejammednetwork theninCIP-1, y ij =1 .Indeed,if i and j areadjacentthenthereexistsasequenceofpairwise adjacentvertices: f ( i 0 ;i 1 ) ;:::; ( i m 1 ;i m ) g ; (4–24) where i 0 = i; and i m = j .Usinginductionitcanbeshownthat y i 0 i k =1 ; 8 k =1 ; 2 ;:::;m From( 4–16 ),wehavethat y i k i k +1 =1 .If y i 0 i k =1 ,thenby( 4–17 ), y i 0 i k +1 y i 0 i k y i k i k +1 =1 whichprovestheinductionstep. TheprovenpropertyimpliesthatinCIP-1: m X j =1 j 6 = i y ij connectivityindexof i: (4–25) 68


Therefore,if ( x ;y ) and ( x ;y ) areoptimalsolutionsofCIP-1andCIPcorrespondingly, then: V ( x ) V ( x ) ; (4–26) where V istheobjectiveinCIP-1andCIP. As ( x ;y ) isfeasibleinCIP,itcanbeeasilycheckedthat y satisesallfeasibility constraintsinCIP-1(itfollowsfromthedenitionof y ij inCIP).So, ( x ;y ) isfeasibleinCIP-1;thusprovingtherststatementofthelemma. HencefromCIP-1, V ( x ) V ( x ) : (4–27) From( 4–26 )and( 4–27 ): V ( x )= V ( x ) : (4–28) Letusdene y suchthat y ij :=1 j isreachablefrom i inthenetworkjammedby x : Using( 4–25 ), ( x ;y ) isfeasibleinCIP-1,andhenceoptimal.Fromtheconstructionof y it followsthat ( x ;y ) isfeasibleinCIP.Relyingon( 4–28 )wecanclaimthat x isanoptimal solutionofCIP.Thelemmaisproved. Wehavethereforeestablishedaone-to-onecorrespondence betweenformulationsCIPandCIP-1.Now,wecanlinearizetheintegerprogramCIP-1byapplyingsomestandard 69


transformations.Theresultinglinear0-1program,CIP-2isgivenas (CIP-2) Minimize n X j =1 c j x j (4–29) s.t. y ij v 0 ij ; 8 i;j =1 ;:::; M ; (4–30) y ij y ik + y kj 1 ;k 6 = i;j ; 8 i;j 2M ; (4–31) v 0 ij v ij + z j + z i 2 ;i 6 = j ; 8 i;j 2M ; (4–32) m X j =1 j 6 = i y ij L; 8 i 2M ; (4–33) M (1 z i ) >S i C i Mz i ; 8 i 2M ; (4–34) z i 2f 0 ; 1 g ; 8 i 2M ; (4–35) x j 2f 0 ; 1 g ; 8 j 2N ;y ij 2f 0 ; 1 g8 i;j 2M ; (4–36) v ij 2f 0 ; 1 g ; 8 i;j 2M ;v 0 ij 2f 0 ; 1 g ; 8 i;j 2M : (4–37) Inthefollowinglemma,weprovideaproofofequivalencebet weenCIP-1andCIP-2. Lemma7. IfCIP-1 hasanoptimalsolutionthenCIP-2 hasanoptimalsolution.Furthermore, anyoptimalsolution x ofCIP-2 isanoptimalsolutionofCIP-1 Proof. For0-1variablesthefollowingequivalenceholds: y ij y ik y kj y ij y ik + y kj 1 TheonlydifferencesbetweenCIP-1andCIP-2aretheconstraints: v 0 ij = v ij z j z i (4–38) v 0 ij v ij + z i + z j 2 (4–39) Notethat( 4–38 )implies( 4–39 )( v ij z j z i v ij + z i + z j 2 ).Therefore,thefeasibilityregionofCIP-2includesthefeasibilityregionofCIP-1.Thisprovestherststatementofthelemma. 70


Fromthelastpropertywecanalsodeducethatforall x 1 ;x 2 suchthat x 1 isanoptimal solutionofCIP-1,and x 2 isoptimalforCIP-2,that V ( x 1 ) V ( x 2 ) ; (4–40) where V ( x ) istheobjectiveofCIP-1andCIP-2. Let ( x ;y ;v 0 ;z ) beanoptimalsolutionofCIP-2.Construct v 00 usingthefollowingrules: v 00 ij := 8>><>>: 1 ; if v ij + z i + z j 2=1 ; 0 ; otherwise : (4–41) v 0 ij v 00 ij ) ( x ;y ;v 00 ;z ) isfeasibleinCIP-2( y ij v 00 ij ),henceoptimal(theobjectivevalueis V ( x ) ,whichisoptimal).Using( 4–41 ), ( v 00 ;z ) satises: v 00 ij = v ij z j z i : Usingthiswehavethat ( x ;y ;v 00 ;z ) isfeasibleforCIP-1.If x 1 isanoptimalsolutionofCIP-1 then: V ( x 1 ) V ( x ) : (4–42) Ontheotherhand,using( 4–40 ): V ( x ) V ( x 1 ) : (4–43) ( 4–42 )and( 4–43 )togetherimply V ( x 1 )= V ( x ) .Thelastequalityprovesthat x isanoptimal solutionofCIP-1.Thus,thelemmaisproved. Wehaveasaresultoftheabovelemmatathefollowingtheorem whichstatesthatthe optimalsolutiontothelinearizedintegerprogramCIP-2isanoptimalsolutiontotheoriginal connectivityindexproblemCIP. Theorem12. IfCIPhasanoptimalsolutionthenCIP-2 hasanoptimalsolution.Furthermore, anyoptimalsolutionofCIP-2 isanoptimalsolutionofCIP. Proof. ThetheoremisanimmediatecorollaryofLemma 6 andLemma 7 71


4.4DeterministicSetupwithPercentileConstraints Aswehaveseen,tosuppresscommunicationonawirelessnetw orkmaynotnecessarily implythatallnodesmustbejammed.Wemightinsteadchooset oconstraintheconnectivity indexofthenodesasintheCIPformulations.Alternatively,itmaybesufcienttojamsom e percentageofthetotalnumberofnodesinordertoacquirean effectivecontroloverthenetwork. Thelattercanbeaccomplishedbyadding percentileriskconstraints tothemathematical formulation.Usedextensivelyinnancialengineeringapp licationsandoptimizationofstochastic systems,riskmeasureshaveproveneffectivewhenappliedt odeterministicproblems[ 120 ].In thissection,wereviewtworiskmeasures,namelyValueatRi sk(VaR)andConditionalValueat Risk(CVaR)andprovideformulationoftheWNJPwiththeincorporationoftheseriskmeasures. 4.4.1Value-at-Risk(VaR)andConditionalValue-at-Risk( CVaR) TheValue-at-Risk(VaR)percentilemeasureisperhapsthem ostwidelyusedinall applicationsofriskmanagement[ 103 ].Statedsimply,VaRisanupperpercentileofagiven lossdistribution.Inotherwords,givenaspeciedconden celevel ,thecorresponding -VaR isthelowestamount suchthat,withprobability ,thelossislessorequalto [ 121 ].VaR typeriskmeasuresarepopularforseveralreasonsincludin gtheirsimpledenitionandeaseof implementation. AnalternativeriskmeasureisConditionalValue-at-Risk( CVaR).DevelopedbyRockafellar andUryasev,CVaRisapercentileriskmeasureconstructedf orestimationandcontrolofrisks instochasticanduncertainenvironments.However,CVaR-b asedoptimizationtechniquescan alsobeappliedinadeterministicpercentileframework.CV aRisdenedastheconditional expectedlossundertheconditionthatitexceedsVaR[ 168 ].Figure 4-2 providesagraphical representationoftheVaRandCVaRconcepts.Aswewillsee,C VaRhasmanypropertiesthat offernicealternativestoVaR. Let f ( x;y ) beaperformanceorlossfunctionassociatedwiththedecisi onvector x X R n ,andarandomvectorin y 2 R m .The y vectorcanbeinterpretedastheuncertaintiesthatmay affecttheloss.Then,foreach x 2 X ,thecorrespondingloss f ( x;y ) isarandomvariablehaving 72


Figure4-2:GraphicalrepresentationofVarandCVaR. adistributionin R whichisinducedby y .Weassumethat y isgovernedbyaprobabilitymeasure P onaBorelset,say Y .Therefore,theprobabilityof f ( x;y ) notexceedingsomethresholdvalue isgivenby ( x; ):= P f y j f ( x;y ) g : (4–44) Foraxeddecisionvector x ( x; ) isthecumulativedistributionfunctionofthelossassocia ted with x .ThisfunctionisfundamentalfordeningVaRandCVaR[ 121 ]. Withthis,the -VaRand -CVaRvaluesforthelossrandomvariable f ( x;y ) forany specied 2 (0 ; 1) aredenotedby ( x ) and ( x ) respectively.Fromtheaforementioned denitions,theyaregivenby ( x ):=min f 2 R : ( x; ) g ; (4–45) and ( x ):= E f f ( x;y ) j f ( x;y ) ( x ) g : (4–46) Noticethattheprobabilitythat f ( x;y ) ( x ) isequalto 1 .Finallybydenition,wehave that ( x ) istheconditionalexpectationthatthelosscorresponding to x isgreaterthanorequal to a ( x ) [ 162 ]. 73


ThekeytoincludingVaRandCVaRconstraintsintoamodelare thecharacterizationsof ( x ) and ( x ) intermsofafunction F : X R 7! R denedby F ( x; ):= + 1 (1 ) E f max f f ( x;y ) ; 0 gg : (4–47) Thefollowingtheorem,whichprovidesthecrucialproperti esofthefunction F followdirectly fromthepaperbyRockafellarandUryasev[ 162 ]. Theorem13. Asafunctionof F ( x; ) isconvexandcontinuouslydifferentiable.The -CVaR ofthelossassociatedwithany x 2 X canbedeterminedfromtheformula ( x )=min 2 R F ( x; ) : (4–48) Inthisformula,thesetconsistingofthevaluesof forwiththeminimumisattained,namely A ( x )=argmin 2 R F ( x; ) ; (4–49) isanonempty,closed,boundedinterval,andthe -VaRofthelossisgivenby ( x )= leftendpointof A ( x ) : (4–50) Inparticular,itisalwaysthecasethat ( x ) 2 argmin 2 R F ( x; ) and ( x )= F ( x; ( x )) : (4–51) Thisresultprovidesanefcientlinearoptimizationalgor ithmforCVaR.However,froma numericalperspective,theconvexityof F ( x; ) withrespectto x and asprovidedbyTheorem 13 ismorevaluablethantheconvexityof ( x ) withrespectto x .Aswewillseeinthefollowing theoremduetoRockafellarandUryasev[ 163 ],thisallowsustominimizeCVaRwithouthaving toproceednumericallythroughrepeatedcalculationsof ( x ) forvariousdecisions x 74


Theorem14. Minimizing ( x ) withrespectto x 2 X isequivalenttominimizing F ( x; ) over all ( x; ) 2 X R ,inthesensethat min x 2 X ( x )=min ( x; ) 2 X R F ( x; ) ; (4–52) wheremoreover ( x ; ) 2 argmin ( x; ) 2 X R F ( x; ) x 2 argmin x 2 X ( x ) ; 2 argmin 2 R F ( x ; ) : (4–53) InthedeterministicsettingoftheWNJP,wearenotparticularlyinterestedinminimizing VaRorCVaRasitpertainstotheloss.Rather,wewouldliketo imposepercentileconstraintson theoptimizationmodelinordertohandleadesiredprobabil itythreshold.Thefollowingtheorem from[ 163 ]providesthiscapability. Theorem15. Foranyselectionofprobabilitythresholds i andlosstolerances i ;i =1 ;:::;m theproblem min x 2 X g ( x ) (4–54) s.t. i ( x ) i ; for i =1 ;:::;m; (4–55) where g isanyobjectivefunctiondenedon X ,isequivalenttotheproblem min ( x; 1 ;:::; m ) 2 X R m g ( x ) (4–56) s.t. F i ( x; i ) i ; for i =1 ;:::;m: (4–57) Indeed, ( x ; 1 ;:::; m ) solvesthesecondproblemifandonlyif x solvestherstproblemand theinequality F i ( x; i ) i holdsfor i =1 ;:::;m Furthermore, i ( x ) i holdsforall i =1 ;:::;m .Inparticular,foreach i suchthat F i ( x ; )= i ,onehasthat i ( x )= i 75


4.4.2PercentileConstraintsandtheWNJP Inthissection,weinvestigatetheuseofVaRandCVaRconstr aintswhenappliedtothe formulationsoftheWNJPderivedinSections 4.3 and 4.4 above.Aswehaveseen,riskmeasures aregenerallydesignedforoptimizationunderuncertainty .Sinceweareconsideringdeterministic formulationsoftheWNJP,wecaninterpreteachcommunicationnode i 2M asarandom scenario,andapplythedesiredriskmeasuresinthiscontex t. WebeginwiththeOPTIMALNETWORKCOVERINGformulationoftheWNJP.Supposeit isdeterminedthatjammingsomefraction 2 (0 ; 1) ofthenodesissufcientforeffectively dismantlingthenetwork.Thiscanbeaccomplishedbytheinc lusionof -VaRconstraintsinthe originalmodel.Let y : M7!f 0 ; 1 g beasurjectiondenedby y i := 8>><>>: 1 ; ifnode i isjammed, 0 ; otherwise : (4–58) RecallfromSection 4.3 that N = f 1 ;:::;n g isthesetoflocationsforthejammingdevices, and x isabinaryvectoroflength n where x j =1 ifajammingdeviceisplacedatlocation j Thentondtheminimumnumberofjammingdevicesthatwillal lowforcovering 100% of thenetworknodeswithprescribedlevelsofjamming C i ,wemustsolvethefollowinginteger program (ONC-VaR) Minimize n X j =1 c j x j (4–59) s.t. m X i =1 y i m; (4–60) n X j =1 d ij x j C i y i ;i =1 ; 2 ;:::;m; (4–61) x j 2f 0 ; 1 g ;j =1 ; 2 ;:::;n; (4–62) y i 2f 0 ; 1 g ;i =1 ; 2 ;:::;m: (4–63) 76


NoticethatthisformulationdiffersfromtheONCformulationwiththeadditionofthe -VaR constraint( 4–60 ).Accordingto( 4–61 ),if y i =1 thennode i isjammed.Lastly,wehavefrom ( 4–60 )thatatleast 100 % ofthe y variablesareequalto 1 TheoptimalsolutiontotheONC-VaRformulationwillprovidetheminimumnumberof jammingdevicesrequiredtosuppresscommunicationonatle ast 100 %ofthenetworknodes. Theresultingsolutionmayprovidecoveragelevelscompara bletothoseprovidedbytheONCmodel,whilepotentiallyreducingthenumberofjammingdev icesused.However,noticethatthe remaining (1 ) 100 %ofthenodesforwhich y i ispotentially 0 ,thereisnoguaranteethatthey willreceiveanyamountofcoverage.Furthermore,theaddit ionofthe m binaryvariablesaddsa computationalburdentoaproblemwhichisalready NP -hard. WecanalsoreformulatetheCONNECTIVITYINDEXPROBLEMtoincludeValue-at-Risk constraints.Let : M7! Z + beasurjectionwhere i returnstheconnectivityindexofnode i .Thatis, i := P mj =1 ;j 6 = i y ij .Furtherlet w : M7!f 0 ; 1 g beadecisionvariablehavingthe propertythatif w i =1 ,then i L .Withthis,theconnectivityindexformulationofWNJPwith VaRpercentileconstraintsisgivenas (CIP-VaR) Minimize n X j =1 c j x j (4–64) s.t. i Lw i +(1 w i ) M;i =1 ; 2 ;:::;m; (4–65) m X i =1 w i m; (4–66) x j 2f 0 ; 1 g ;j =1 ; 2 ;:::;n; (4–67) w i 2f 0 ; 1 g ;i =1 ; 2 ;:::;m; (4–68) i 2f 0 ; 1 g ;i =1 ; 2 ;:::;m; (4–69) where M 2 R issomelargeconstant. Analogoustoconstraints( 4–60 )-( 4–61 ),constraints( 4–65 )-( 4–66 )guaranteethatat least 100% ofthenodeswillhaveconnectivityindexlessthan L .AswiththeONC-VaR77


formulation,therearetwodrawbacksofCIP-VaR.First,thereisnocontrolguaranteeatallonany oftheremaining (1 ) 100% nodesforwhich w i =0 .Secondly,theadditionof m binary variablesaddsatremendouscomputationalburdentothepro blem.AsanalternativetoVaR,we nowexamineformulationsoftheWNJPusingConditionalValue-at-Riskconstraints[ 162 ]. WerstconsidertheOPTIMALNETWORKCOVERINGproblem.Inordertoputthisintoour derivedframework,weneedtodenethelossfunctionassoci atedwithaninstanceoftheONC. Weintroducethefunction f : f 0 ; 1 g n M7! R denedby f ( x;i ):= C i n X j =1 x j d ij : (4–70) Thatis,givenadecisionvector x representingtheplacementofthejammingdevices,theloss functionisdenedasthedifferencebetweentheenergyrequ iredtojamthenetworknode i and thecumulativeamountofenergyreceivedatnode i dueto x .Withthis,wecanformulatetheONCwiththeadditionofCVaRconstraintsasthefollowinginteg erlinearprogram: (ONC-CVaR) Minimize n X j =1 c j x j (4–71) s.t. + 1 (1 ) m m X i =1 max C min n X j =1 x j d ij ; 0 0 ; (4–72) 2 R ; (4–73) x j 2f 0 ; 1 g ; (4–74) where C min istheminimalprescribedjammingleveland d ij isdenedasabove.Theexpression onthelefthandsideof( 4–72 )is F ( x; ) .Further,fromTheorem 15 weseethatconstraint ( 4–72 )correspondstohaving ( x ) =0 [ 163 ].Saiddifferently,theCVaRconstraint ( 4–72 )impliesthatinthe (1 ) 100% oftheworst(least)coverednodes,theaveragevalueof f ( x ) 0 .Forthecasewhen C i C forall i ,itfollowsthattheaveragelevelofjammingenergy receivedbytheworst (1 ) 100% ofnodesexceeds C 78


Theimportantpointaboutthisformulationisthatwehaveno tintroducedadditionalinteger variablestotheprobleminordertoaddthepercentileconst raints.Recall,thatinONC-VaRweintroduced m discretevariables.Sincewehavetoaddonly m realvariablestoreplace max -expressionsunderthesummationandarealvariable ,thisformulationismucheasierto solvethanONC-VaR. Inasimilarmanner,wecanformulatetheCONNECTIVITYINDEXPROBLEMwiththe additionofCVaRconstraints.Asbefore,weneedtorstden eanappropriatelossfunction. Recallthatthedenitionof i ,theconnectivityindexofnode i ,isgivenasthenumberofnodes reachablefrom i .Thencandenethelossfunction f 0 foranetworknode i asthedifference betweentheconnectivityindexof i andthemaximumallowableconnectivityindex L which occursasaresultoftheplacementofthejammingdevicesacc ordingto x .Thatis,let f 0 : f 0 ; 1 g n M7! Z bedenedby f 0 ( x;i ):= i L: (4–75) Withthis,theCIP-CVaRformulationisgivenasfollows. (CIP-CVaR) Minimize n X j =1 c j x j (4–76) s.t. + 1 (1 ) m m X i =1 max f i L ; 0 g 0 ; (4–77) i 2 Z ; (4–78) 2 R ; (4–79) where i isdenedasabove.Aswiththepreviousformulation,theexp ressionontheleft-hand sideof( 4–77 )is F ( x; ) from( 4–47 ).Furthermore,wehavefromfromTheorem 15 that ( 4–77 )correspondstohaving ( x ) =0 .ThisconstraintonCVaRprovidesthatforthe (1 ) 100% oftheworstcases,theaverageconnectivityindexwillnote xceed L .Again,we seethatinordertoincludetheCVaRconstraint,weonlyneed toadd ( m +1) realvariablestothe 79


Table4-1:Optimalsolutionsusingthecoverageformulatio nwithregularandVaRconstraints. OptimalSolutions RegularConstraints VaRConstraints NumberofJammers 6 4 LevelofJamming 100% 8 nodes 100% for 96% ofnodes, 85%(ofreqd.)for4%ofnodes CPLEX R r Time 0.81sec 0.98sec problem.Computationally,CVaRprovidesamoreconservati vesolutionandwillbemucheasier tosolvethantheCIP-VaRformulationaswewillseeinthenextsection. 4.5CaseStudiesandAlgorithms Inordertodemonstratetheadvantagesanddisadvantagesof theproposedformulationsfor theWNJP,wewillpresenttwocasestudies.Theexperimentswereperf ormedonaPCequipped witha1.4MHzIntelPentium R r 4processorwith1GBofRAM,workingundertheMicrosoft Windows R r XPSP1operatingsystem.Intherststudy,anexamplenetwor kisgivenandthe problemismodeledusingtheproposedcoverageformulation .Theproblemisthensolvedexactly usingthecommercialintegerprogrammingsoftwarepackage ,CPLEX R r .Next,wemodifythe problemtoincludeVaRandCVaRconstraintsandagainuseCPL EX R r tosolvetheresulting problems.Numericalresultsarepresentedandthethreefor mulationsarecompared.Inthe secondcasestudy,wemodelandsolvetheproblemusingtheco nnectivityindexformulation.We thenincludepercentileconstraintsre-optimize.Finally ,weanalyzetheresults. 4.5.1CoverageFormulation HerewepresenttwonetworksandsolvetheWNJPusingthenetworkcovering(ONC) formulation.Therstnetworkhas100communicationnodesa ndthenumberofavailable jammingdevicesis36.Thecostofplacingajammingdeviceat location j c j isequalto1forall locations.Thisproblemwassolvedusingtheregularconstr aintsandtheVaRtypeconstraints. Recallthatthereisasetofpossiblelocationsatwhichjamm ingdevicescanbeplaced.Inthese examples,thissetofpointsconstitutesauniformgridover thebattlespace.Theplacementofthe jammingdevicesfromeachsolutioncanbeseeninFigure 4-3 .Thenumericalresultsdetailing thelevelofjammingforthenetworknodesisgiveninTable 4-1 .NoticethattheVaRsolution 80


Figure4-3:Casestudy1.Theplacementofjammersisshownwh entheproblemissolvedusing theoriginalandVaRconstraints.Table4-2:Optimalsolutionsusingthecoverageformulatio nwithregularandVaR,andCVaR constraints. OptSolns Reg(all) VaR(.9conf) CVaR(.7conf) #Jammers 9 8 7 JammingLevel 100% 8 nodes 100% for 90% ofnodes,72%for10%ofnodes 100%for57%ofnodes,90%for20%ofnodes,76%for23%ofnodes CPLEX R r Time 15sec 15h55min11sec 41sec calledfor33%lessjammingdevicesthantheoriginalproble mwhileprovidingalmostthesame jammingquality. Inthesecondexample,thenetworkhas100communicationnod esand72available jammers.Thisproblemwassolvedusingtheregularconstrai ntsaswellasbothtypesof 81


Figure4-4:Casestudy1continued.Theplacementofjammers isshownwhentheproblemis solvedusingVaRandCVaRconstraints.percentileconstraints.TheresultinggraphisshowninFig ure 4-4 .Thecorrespondingnumerical resultsaregiveninTable 4-2 Inthisexample,theVaRformulationrequires11%lessjammi ngdeviceswithalmostthe samequalityastheformulationwiththestandardconstrain ts.However,thisformulationrequires nearly16hoursofcomputationtime.TheCVaRformulationgi vesasolutionwithaverygood jammingqualityandrequires22%lessjammingdevicesthant hestandardformulationand11% lessdevicesthantheVaRformulation.Furthermore,theCVa Rformulationrequiresanorderof magnitudelesscomputingtimethantheformulationwithVaR constraints. 4.5.2ConnectivityFormulation WenowpresentacasestudywheretheWNJPwassolvedusingtheconnectivityindex formulation(CIP).Thecommunicationgraphconsistsof30nodesand60edges. Themaximal numberofjammingdevicesavailableis36.Wesetthemaximal allowedconnectivityindexof anynodetobe3.InFigure 4-5 wecanseetheoriginalgraphwiththecommunicationlinkspr ior tojamming.TheresultoftheVaRandCVaRsolutionsisseenin Figure 4-6 .Thecondence 82


Figure4-5:CaseStudy2:Originalgraph. (a) (b) Figure4-6:Acomparisonofthepercentileconstrainedsolu tions.Inbothcases,thetriangles representtheplacementofjammingdevices.(a)VaRSolutio n.(b)CVaRSolution. levelforboththeVaRandCVaRformulationswas0.9.Bothfor mulationsprovideoptimal solutionsforthegiveninstance.Theresultingcomputatio ntimefortheVaRformulationwas15 minutes34seconds,whiletheCVaRformulationrequiredonl y7minutes33seconds. 83


4.6ConcludingRemarks InthischapterweintroducedthedeterministicWIRELESSNETWORKJAMMINGPROBLEMandprovidedseveralformulationsusingnodecoveringcons traintsaswellasconstraintsonthe connectivityindicesofthenetworknodes.Wealsoincorpor atedpercentileconstraintsintothe derivedformulations.Further,weprovidedtwocasestudie scomparingthetwoformulationswith andwithouttheriskconstraints. Withtheintroductionofthisproblem,wealsorecognizetha tseveralextensionscanbe made.Forexample,alloftheformulationspresentedinthis chapterassumethatthenetwork topologyoftheenemynetworkisknown.Itisreasonabletoas sumethatthisisnotalwaysthe case.Infact,theremaybelittleornoaprioriinformationa boutthenetworktobejammed. Inthiscase,stochasticformulationsshouldbeconsidered andanalyzed.Thisbringsustothe topicofthenextchapterinwhichweconsiderthecasewhenno informationisknownaboutthe networktobejammedotherthanitsrelativelocationinside aplanarregion. 84


CHAPTER5 JAMMINGCOMMUNICATION NETWORKSUNDERCOMPLETEUNCERTAINTY 5.1Introduction Thischapterdescribesaproblemofinterdicting/jammingc ommunicationnetworks inuncertainenvironments[ 44 ].Interdictionofcommunicationnetworksisanimportant application,butaspreviouslymentioned,hasnotbeeninte nsivelyresearcheddespitethevast amountofworkonoptimizingtelecommunicationsystems[ 155 ].Mostpapersonnetwork interdictionareaboutpreventingjammingandanalyzingne tworkvulnerability[ 68 134 ].To ourknowledge,theonlyliteratureonnetworkinterdiction involvingoptimalplacementof jammingdevicesistheworkofCommanderetal.[ 45 ](presentedinChapter 4 )inwhichseveral mathematicalprogrammingformulationsweregivenforthed eterministicWIRELESSNETWORK JAMMINGPROBLEM.Theonlyotherthoroughlystudiedcasesareproblemsofmin imizingthe maximalnetworkowandmaximizingtheshortestpathbetwee ngivennodesviaarcinterdiction usinglimitedresources.Cormicanetal.[ 49 ],Israelietal.[ 110 ],andWood[ 174 ]studied stochasticanddeterministiccasesandsuggestedefcient heuristics.Asimilarsetupbutwitha differentobjectivewasrecentlystudiedbyHeldin 2005 [ 95 ]. Thisproblemisparticularlyimportantintheglobalwaront errorismasimprovised explosivedevices(IEDs)continuetoplaguethecoalitionf orcesinIraq.Thesehomemade bombsarealmostalwaysdetonatedbysomeformofradiofrequ encydevicesuchascellular telephones,pagers,andgaragedooropeners.Theabilityto suppressradiowavesinaregion willhelppreventcasualtiesresultingfromIEDs.Furtherm ore,sincemostsituationsarisein militarybattleeldscenarios,exactinformationaboutth etopologyoftheadversary'snetwork isunknown.Thus,deterministicnetworkinterdictionappr oacheshavelimitedapplicability.In thiscase,astochasticapproachinvolvingsomeriskmeasur eforevaluatingtheefciencyofthe jammingdeviceplacementmaybehelpful.However,choosing anappropriateriskmeasureisa challengingprobleminitsownright.Inthischapter,wecon sideranextremecasewherethereis 85


noaprioriinformationaboutthetopologyofthenetworktob ejammed.Theonlyinformation usedinourapproachisaboundingarea,containingthecommu nicationnetwork. Theorganizationofthischapterisasfollows.Section 5.3 givesaformaldescriptionof theproblemandthejammingmodel.Wederiveboundsandprove aconvergenceresultforthe caseofcompleteuncertaintyinSection 5.4 .Herewealsodemonstratetheadvantageofthe proposedmethodcomparedtothesimpliedcasewhichdoesno taccountforthecumulative effectofthejammingdevices.InSection 5.5 wepresentarandomizedlocalsearchandillustrate itseffectivenessbyusingtheboundsderivedinthepreviou ssection.Section 5.6 providessome concludingremarks. 5.2Descriptions,Assumptions,andDenitions Ingeneral,theproblemofjammingacommunicationnetworki stodeterminetheminimum numberofjammingdevicesrequiredtointerdictorsuppress functionalityofthenetwork. Startingwiththisgeneralstatement,morespeciconescan beobtainedbyconsideringvarious typesofjammingdevicesandinterdictioncriteria.Depend ingonthegiveninformationabout thecommunicationnodesandthenetworktopology,stochast icordeterministicsetupscan beconstructed[ 45 ].Belowweprovideassumptionsandbasicdenitionsofthec onsidered framework. Weconsiderradio-transmittingcommunicationnetworksan djammingdevicesoperating withelectromagneticwaves.Weassumethatthejammingdevi ceshaveomnidirectionalantennas andemitelectromagneticwavesinalldirectionswiththesa meintensity.Wealsoassumethat jammingpowerdecreasesreciprocallytothesquareddistan cefromadevice. Denition12. Apoint(communicationnode) X issaidtobejammedorcoveredifthe cumulativeenergyreceivedfromalljammingdevicesexceed ssomethresholdvalue E : X i R 2 ( X;i ) E; (5–1) 86


where 2 R and R ( X;i ) representsthedistancefrom X tojammingdevice i .Thiscondition canberewrittenas: X i 1 R 2 ( X;i ) 1 L 2 ; (5–2) where L = q E Thelatterinequalityimpliesthatajammingdevicecoversa nypointinsideacircleofradius L Denition13. Aconnection(arc)betweentwocommunicationnodesisconsi deredblockedifany ofthetwonodesiscovered. Usually,interdictionefciencyisdeterminedbyafractio nofcoverednodesand/orarcs. Morecomplicatedcriteriausedarebasedontheamountofinf ormationtransmittedthroughthe networkorthelengthoftheshortestpathbetweenpairsofno des.Wedonotconsideraspecic criteriumbecauseweareinterestedinthecaseofcompleteu ncertainty.Thus,weareassuming thatwehavenoknowledgeofthenetworktopology,including informationaboutthenode coordinates. 5.3ProblemFormulation Ifweignorethecumulativeeffectofthejammingdevices,th entheproblemreducesto determiningtheoptimalcoveringofanareaonaplanebycirc les.Thiscoveringproblemwas solvedin 1936 byKershner[ 113 ].Thecurrentchaptershowsthataccountingforthecumulat ive effectofallthedevicescanleadtosignicantlossesincos ts,i.e.requirednumberofjamming devices. Sinceweassumenoinformationisknownaboutthenetworktob ejammed,theonly reasonableapproachistocoverallpointsinsomeareaknown tocontainthenetwork.This approachwouldalsobeappropriatewhensomeinformationab outthenetworkisavailable,butis potentiallyinaccurate. Weconsideracasewhenacommunicationnetworkislocatedin sideasquare.However, allofthefollowingtheoremscanbeformulatedforamoregen eralcase.Forexample,to 87


Figure5-1:Uniformgridwithjammingdevices obtainresultswhenthenetworkiscontainedinsidearectan gularregionintheplane,theonly modicationrequiredtothecalculationsisanappropriate updatingofthesummationbounds. Anoptimalcoveringisonewhichcontainstheminimumnumber ofjammingdevicesthat jamallpointsintheparticularareaofinterest.However, ndingagloballyoptimalsolutionfor thegeneralproblemisdifcult[ 45 ].Therefore,weconsiderasubproblemofcoveringasquare withjammingdeviceslocatedatthenodesofauniformgrid.T hesolutiontothisproblemwill provideafeasiblesolution(optimalincertaincases)toth egeneralproblem.Supposethegrid stepsizeis R .Ifthelengthofasquareside a isnotamultipleof R ,thenwecoverabigger squarewithasideoflength R ([ a R ]+1) .SeeFigure 5-1 foranexample.Theoptimalsolution intheconsideredproblemisauniformgridwiththelargestp ossiblestepsizewhichcoversthe square.Theproblemremainsnon-trivial,evenforthissimp liedsetup. Lemma8. Foranycoveringofasquarewithauniformgrid,apointwhich receivestheleast amountofjammingenergyliesinsideacornergridcell(seeF igure 5-2 ). Proof. Consideracornercell S 0 andanarbitrarynon-cornercell S i .Weprovethatforanypoint P 2 S i ,thereisacorrespondingpoint P 0 2 S 0 suchthat E ( P ) >E ( P 0 ) ,where E ( X ) isthe cumulativejammingenergyfromalldevicesreceivedatpoin t X Let P 0 beasymmetriccorrespondenceofpoint P inside S 0 .Here,symmetryimpliesthat P and P 0 areequidistantfromthesidesoftheirrespectivecells.We splitthesquareintothe 88


Figure5-2:Theleastcoveredpointisshowninthelowerleft gridcell. fourrectangles A;B;C; and D ,where A istherectanglecontainingcells S 0 and S i (seeFigure Figure5-3:SquareDecomposition 5-3 ).Denotetheothertwocornercellsofrectangle A by C 1 and C 2 .Letalso T 1 and T 2 bepoints inside C 1 and C 2 respectively,suchthat T 1 PT 2 P 0 isarectanglewithsidesparalleltothesidesof thesquareasinFigure 5-4 .Usingsymmetrywegetthefollowingrelations: E ( P 0 ;A )= E ( P;A ) ; (5–3) E ( P 0 ;B )

Figure5-4:EquivalentPoints where E ( X;I ) isthecumulativejammingenergyfromalldevicesinsiderec tangle I receivedby point X .Relations( 5–3 )-( 5–6 )imply E ( P 0 )= E ( P 0 ;A )+ E ( P 0 ;B )+ E ( P 0 ;C )+ E ( P 0 ;D )

Figure5-5:Cumulativeemanationofjammingdevices. bottomleftcornercellasshowninFigure 5-5 I 1 ;I 2 ,and I 3 arecumulativejammingenergy receivedat P byjammingdeviceslocatedinregions C;A ,and B correspondingly.Similarly, I 4 isthejammingenergyfromthejammingdevicelocatedattheb ottomleftnode O .Withthis,the jammingenergyreceivedatpoint P iscalculatedthroughtheexpression E ( P )= I 1 + I 2 + I 3 + I 4 ; where(5–9) I 1 = T 1 X i =0 T 1 X j =0 1 ( R x 0 + i R ) 2 +( R y 0 + j R ) 2 ; (5–10) I 2 = T 1 X i =0 1 ( R x 0 + i R ) 2 + y 2 0 ; (5–11) I 3 = T 1 X j =0 1 x 20 +( R y 0 + j R ) 2 ; (5–12) I 4 = 1 x 20 + y 2 0 ; (5–13) T = h a R i +1 : (5–14) Noticethatwecanestimate I 2 + I 3 as I 2 + I 3 2 T 1 X i =0 1 R 2 (1+ i ) 2 + R 2 2 R 2 Z T 0 1 1+(1+ x ) 2 dx: (5–15) 91


Figure5-6:IntegralLowerBound. Thisfollowsfromthefactthat N X i =0 f ( i ) Z N +1 0 f ( x ) dx; (5–16) where f ( x ) isadecreasingfunction.Thispropertycanbeeasilyestabl ishedgeometrically. NoticeinFigure 5-6 thattheleftsideofinequality( 5–16 )representstheshadedregioninthe gure,whiletherightsiderepresentstheareaunder f ( x ) .Continuingfrom( 5–15 )abovewe have Z T 0 1 1+(1+ x ) 2 dx =arctan( T +1) 4 = 2 arctan 1 T +1 4 (5–17) 4 1 T +1 : Hereandfurther,weusetheinequalitiesgivenbelow: arctan( x ) x; 0 x 1 ; (5–18) arctan( x ) x x 3 3 ; 0 x 1 : (5–19) 92


Nowcombining( 5–15 )and( 5–17 ),weobtain I 2 + I 3 2 R 2 4 1 T +1 : (5–20) Wealsohavethefollowingapproximationfor I 4 whichfollowsclearly I 4 1 2 R 2 : (5–21) Forestimating I 1 weuseapropertysimilarto( 5–16 ),butinahigherdimension.Namely, N X i =0 N X j =0 f ( i;j ) Z N +1 0 Z N +1 0 f ( x;y ) dxdy; (5–22) whereasabove, f ( x;y ) isadecreasingfunctionof x and y .Usingthisinequality,wederivethe followingapproximationfor I 1 I 1 Z T 0 Z T 0 dxdy ( R x 0 + x R ) 2 +( R y 0 + y R ) 2 Z T 0 Z T 0 dxdy ( R + x R ) 2 +( R + y R ) 2 (5–23) = 1 R 2 Z T +1 1 Z T +1 1 dxdy x 2 + y 2 : Furthermore, Z T +1 1 Z T +1 1 dxdy x 2 + y 2 = Z T +1 1 1 x arctan T +1 x dx Z T +1 1 1 x arctan 1 x dx Z T +1 1 1 x arctan T +1 x dx Z T +1 1 dx x 2 = Z T +1 1 1 x x arctan x T +1 dx 1+ 1 T +1 (5–24) = 2 ln( T +1) 1+ 1 T +1 Z T +1 0 1 x arctan x T +1 dx 2 ln( T +1) 1+ 1 T +1 Z T +1 0 1 x x T +1 dx = 2 ln( T +1) 2 1 1 T +1 : 93


Combiningthisresultwith( 5–23 )wehave I 1 1 R 2 2 ln( T +1) 2 1 1 T +1 : (5–25) Summing( 5–15 ),( 5–21 ),and( 5–25 )weobtainanoverestimateofthetotalcoverageat point P .Thatis E ( P ) 1 R 2 2 ln( T +1) 2+ 2 T +1 + 2 2 T +1 + 1 2 = 1 R 2 2 ln( T +1)+ 2 3 2 (5–26) 1 2 R 2 ln a R +1 + 3 : Toguaranteecoverageofpoint P ,itissufcienttoclaimthat f ( R )= 1 2 R 2 ln a R +1 + 3 1 L 2 : (5–27) Since f ( R ) ismonotonicallydecreasingon (0 ; + 1 ) ,thelargest R satisfyingtheaboveinequality istheuniquesolution R oftheequation f ( R )= 1 L 2 : (5–28) Thus,auniformgridwithstepsize R jamsanypoint P insideacornercell.AccordingtoLemma 8 ,thegridjamstheleastcoveredpointinthesquareimplying thatthewholesquareisjammed. Thuswehavethedesiredresult. Sincethefunction f ( R )= 1 2 R 2 ( ln( a R +1)+ 3) ismonotonic,equation( 5–8 )canbe easilysolvedusinganumericalproceduresuchasabinaryse arch.Therefore,using( 5–8 ),wecan obtainastepsize R suchthatthecorrespondinguniformgridcoverstheentires quare.Further,it iseasytoseethatthenumberofjammingdevicesinthegriddo esnotexceed N 1 = a R +2 2 : (5–29) 94


Table5-1:Comparing N 2 N 1 forvariousvaluesof k k x N 2 N 1 10 2 2.44 2.3 10 4 3.54 4.8 10 6 4.40 7.5 10 8 5.14 10.2 Amorestraightforwardsolutionoftheinitialproblemcoul dbebasedonthepropertythata jammingdevicecoversallthepointsinsideacircleofradiu s L asmentionedinDenition 12 Usingthat,wecouldreducetheproblemtondingtheoptimal coveringofasquarewithcircles ofradius L .Adirectresultfrom[ 113 ](thatwasmentionedin[ 134 ])isthatinthelimit,the minimumnumberofcirclestocoveranarea a 2 is N 2 = 2 a 2 3 p 3 L 2 : (5–30) Tocomparetheapproaches,weconsidertheratio N 2 N 1 = R L 2 2 3 p 3 1 (1+2 R a ) 2 = 2 x 2 3 p 3 1 (1+ 2 x k ) 2 ; (5–31) where x = R L and k = a L .Usingthesesubstitutions,equation( 5–8 )canberewrittenintermsof variables x and k asfollows 1 x 2 ln k x +1 + 3 =2 : (5–32) Bysolving( 5–32 )fordifferentvaluesof k ,onecanndcorrespondingvaluesof x and N 2 N 1 Toevaluatetheadvantageoftheuniformgridapproachovert henaiveone,weprovidesome computationalresultsintheTable 5-1 .Fromthetable,weseethatas k increases,theadvantage ofusingourapproachbecomesmoresignicant.Infact,itca nbeprovedthat lim a !1 N 2 N 1 = 1 : ThiswillfollowasacorollaryofTheorem 18 Toestablishthequalityofthelowerboundrigorously,wene edtorstestablishasimilar resultforanupperbound.Thisfollowsinthenexttheorem. 95


Theorem17. Theuniquesolutionoftheequation 1 R 2 2 ln 2 a R +1 1 6( a R +1) + 2 + 19 3 = 1 L 2 (5–33) isanupperbound R oftheoptimalgridstepsize R Proof. Let P ( x 0 ;y 0 ) betheleastjammedpoint,thatliesinsideacornercellacco rdingtoLemma 8 .Withoutthelossofgenerality,asintheproofofTheorem 16 ,weassumethat P isinside thebottomleftcornercell.Thejammingenergyreceivedatp oint P iscalculatedthroughthe expressions( 5–9 )-( 5–14 ).Since P istheleastcoveredpoint,thefollowinginequalityholds. E ( P ) E P 0 x = R 2 ;y =0 = I 0 1 + I 0 2 + I 0 3 + I 0 4 ; where(5–34) I 0 1 = T 1 X i =0 T 1 X j =0 1 ( R 2 + i R ) 2 +( R + j R ) 2 ; (5–35) I 0 2 = T 1 X i =0 1 ( R 2 + i R ) 2 ; (5–36) I 0 3 = T 1 X j =0 1 ( R 2 ) 2 +( R + j R ) 2 ; (5–37) I 0 4 = 1 ( R 2 ) 2 : (5–38) I 0 2 and I 0 3 canbeestimatedthroughintegralssimilarlytothetechniq uesusedintheproofof Theorem 16 .Thefollowinginequalityholds N X i =1 f ( i ) Z N 0 f ( x ) dx; (5–39) where f ( x ) isadecreasingfunction.Thispropertycanalsobeprovenge ometrically.Figure 5-7 representsagraphicalinterpretationofthisrelation.Th eleftsideoftheinequalityisrepresented bytheshadedarea.Therightsideof( 5–39 )istheareaunder f ( x ) .Withthispropertywehave 96


Figure5-7:IntegralUpperBound. from( 5–36 )that I 0 2 1 ( R 2 ) 2 + Z T 1 0 dx ( R 2 + x R ) 2 = 1 R 2 6 1 T 1 2 : (5–40) Furthermore,usinginequalities( 5–18 )and( 5–19 ),weseethat( 5–37 )isestimatedby I 0 3 1 ( R 2 ) 2 +( R + x R ) 2 = 2 3 R 2 + 2 R 2 arctan 1 2 arctan 1 2 T 2 3 R 2 + 2 R 2 1 2 1 2 T + 1 24 T 3 (5–41) = 1 R 2 5 3 1 T + 1 12 T 3 : Toestimate I 0 1 apropertysimilarto( 5–39 )canbeused.Thisinequalityisgivenby N X i =1 N X j =1 f ( i;j ) Z N 0 Z N 0 f ( x;y ) dxdy + Z N 0 f ( x; 0) dx + Z N 0 f (0 ;y ) dy; (5–42) 97


where f ( x;y ) isadecreasingfunctionof x and y .Withtheaboveinequality, I 0 1 1 ( R 2 2 )+ R 2 + Z T 1 0 dx ( R 2 ) 2 +( R + x R ) 2 + Z T 1 0 dx ( R 2 + x R ) 2 + R 2 + + Z T 1 0 Z T 1 0 dxdy ( R 2 + x R ) 2 +(( R + y R ) 2 = 4 5 R 2 + C R 2 + 1 R 2 Z T 1 0 Z T 1 0 d ( x + 1 2 ) dy ( 1 2 + x ) 2 +( y +1) 2 ; where(5–43) C =2arctan(2 T ) arctan(2)+arctan T 1 2 2 = 2 2arctan 1 2 T +arctan 1 2 arctan 2 2 T 1 (5–44) 2 2 1 2 T 1 24 T 3 + 1 2 2 2 T 1 8 3(2 T 1) 3 +1 2 : Thedoubleintegralin( 5–43 )isboundedasfollows Z T 1 0 Z T 1 0 d ( x + 1 2 ) dy ( 1 2 + x ) 2 +( y +1) 2 = Z T 1 2 1 2 Z T 1 dtdy t 2 + y 2 = Z T 1 2 1 2 1 t arctan T t arctan 1 t dt Z T 1 2 1 2 1 t 2 arctan t T dt Z T 1 2 1 2 1 t 1 t 1 3 t 3 dt (5–45) 2 ln T 1 2 ln 1 2 Z T 1 2 1 2 1 t t T t 3 3 T 3 dt 4 3 1 T 1 2 + 1 6( T 1 2 ) 2 = 2 ln(2 T 1) 20 3 + 5 6 T + 1 12 T 2 1 36 T 3 + 1 T 1 2 1 6( T 1 2 ) 2 < 2 ln(2 T 1) 20 3 + 5 6 T + 1 T 1 2 1 12( T 1 2 ) 2 : Combiningtheresultsfrom( 5–43 ),( 5–44 ),and( 5–45 )givestheoverestimatefor I 0 1 as I 0 1 < 1 R 2 2 ln(2 T 1)+ 2 16 3 + 5 6 T + 1 T 1 2 1 12( T 1 2 ) 2 : (5–46) 98


Recallequation( 5–34 )stated E ( P ) I 0 1 + I 0 2 + I 0 3 + I 4 .Sousingtheexpressionfor I 0 4 given in( 5–38 )andtheoverestimatesfor I 0 1 ;I 0 2 ,and I 0 3 derivedinequations( 5–46 ),( 5–40 ),and( 5–41 ) respectively,weobtain E ( P ) 1 R 2 2 ln(2 T 1) 1 6 T + 2 + 19 3 : (5–47) Finally,ifwelet T =[ a R ]+1 a R +1 ,weget E ( P ) < 1 R 2 2 ln 2 a R +1 1 6( a R +1) + 2 + 19 3 (5–48) Thefunction f ( R )= 1 R 2 2 ln 2 a R +1 1 6( a R +1) + 2 + 19 3 ismonotone,hencetheequation f ( R )= 1 L 2 hasauniquesolution R .Equation( 5–48 )impliesthatagridwithstepsize R does notcovertheentiresquare.Thatis,thereexistsatleaston epoint P thatremainsuncovered. Thus R isanupperboundfortheoptimalgridcoveringproblem.Sinc etheoptimalgridstepsize R < R ,thetheoremisproved. InFigure 5-8 ,weseeanexampleinwhichwearecoveringat 40 40 squareandthe requiredjamminglevelateachpointis 3 : 0 units.Inpart(a),weseethecoverageassociatedwith therequirednumberofdevicesfromthelowerboundofTheore m 16 .Inthiscase, 20 2 =400 jammingdevicesareusedtocoverthearea.Noticethatthere arenoholesintheregion.This, togetherwiththescallopshelloutsidetheboundingboxind icatesthatallpointswithintheregion arecovered.Inpart(b),weseethecoveragecorrespondingt otheplacementofthejamming devicesonauniformgridaccordingtotheupperboundofTheo rem 17 .Here,therequired numberofdevicesis 19 2 =361 .Noticetheholeslocatedatthefourcornersoftheregion indicatingthatthesepointsareuncovered.Thisvalidates thetheoreticalresultsobtainedin Theorem 16 andTheorem 17 Nowthatwehaveestablishedbothupperandlowerboundsfora noptimalgridstepsize,we candeterminethequalityofthebounds.Theresultisobtain edinthefollowingtheorem. Theorem18. lim a !1 R R =1 ; (5–49) 99

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-25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b) Figure5-8:Comparisonofthelowerandupperbounds.(a)The coverageofwhenjamming devicesareplacedaccordingtothelowerboundfromTheorem 2.Thetotalnumberofjamming devicesrequiredis 20 2 =400 .(b)Weseethecoverageassociatedwiththeresultobtained from Theorem 17 .Inthiscase, 19 2 =361 devicesareplaced.Noticethecornerpointsarenotjammed. where R and R areboundsobtainedfromequations( 5–8 )and( 5–33 ),correspondingly. Moreover,thefollowinginequalityholds: 1 R R r 1+ c ln( a ) ; (5–50) forconstants M 2 R ;c 2 R ,suchthat R>M Proof. Byletting x := R L and y := R L ,equations( 5–8 )and( 5–33 )canberespectivelyrewrittenas a = L x e 2 ( x 2 + 3 2 ) 1 1 ,and(5–51) 2 ln 2 a L y +1 = y 2 19 3 2 + L y 6( a + L y ) : (5–52) Toprovethetheorem,weneedtoshowthat lim a !1 y x =1 ; (5–53) 100

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where x> 0 and y> 0 aresolutionsof( 5–51 )and( 5–52 ),correspondingly.From( 5–52 ),we obtain 2 ln 2 a L y +1 >y 2 C 1 ; where(5–54) C 1 = 19 3 + 2 ; and(5–55) a> L y 2 e 2 ( y 2 C 1 ) 1 : (5–56) From( 5–51 )and( 5–56 )weseethat x e 2 ( x 2 + C 2 ) C 3 1 > y 2 e 2 ( y 2 C 1 ) 1 ; where(5–57) C 2 = 3 2 ; and(5–58) C 3 = e 1 : (5–59) Since y L and x L areupperandlowerbounds,correspondingly,thefollowing relationholds y x > 1 : (5–60) With( 5–51 )and( 5–60 )above,wecanalsoconcludethat lim a !1 x = 1 and lim a !1 y = 1 : (5–61) Forall M 2 R ,where M> p C 1 ,thereexists Q 2 R suchthat( 5–57 )canbereducedto y x M: (5–62) Moreover,for c = 2 ln( Q ) thefollowinginequalityholds y x 2 1 c x 2 ; and y>M: (5–63) Assumeforthesakeofcontradictionthattheinequalityin( 5–63 )doesnotholdforsome ( x ;y ) : Thatisassumethat y x 2 1 > c x 2 .Using( 5–62 )wehave y x
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whichcontradicts( 5–60 ). Applying( 5–60 )and( 5–63 )weget 1 < y x r 1+ c x 2 ; and y>M: (5–65) Letting a tendto 1 andtaking( 5–61 )intoaccount,weseethatinfact lim a !1 y x =1 : (5–66) Finally,byusing( 5–65 )and( 5–51 ),thefollowingrelationcanbeobtained 1 < y x s 1+ k ln( a ) ; (5–67) forsomeconstant k 2 R ,when y>M .Thus,thetheoremisproved. 5.5HeuristicforUncertainJamming Here,wedescribetheimplementationofarandomizedlocals earchheuristicforthecase ofjammingundercompleteuncertainty.Recallthatthesubp roblemforwhichthebounds inTheorem 16 andTheorem 17 wederivedplace n jammingdevices,where n isaperfect square.Theobviousdrawbackofthistechniqueisthesituat ionwhereforexampleR requires25 jammingdevicesand R callsfor16,andtheoptimalsolutiontothegeneralproblem is18.Using theuniformgridapproachwillrequirenearly 40% moredevicesthanareneededtocoverthe region. Pseudo-codeforthelocalsearchisgiveninFigure 5-9 .Theheuristictakesasinputthe sizeoftheregioncontainingthenetwork( region ).Thenumberofjammingdevicesrequiredto covertheareaof region bythelowerboundonthegridstep(upperboundonjammingdev ices) derivedinTheorem 16 ( ubJammers ) isthesecondinputparameter.Inline1,theoptimalsolutio n ( X )issetto ubJammers The while loopfromlines2-12iswherethelocaloptimizationtakespl ace.Inline3,the jammingdevicesarerandomlyscatteredwithinthesquarere gionknowntocontainthenetwork. Nextinthe while loopfromlines5-8,thosepointswhicharereceivingthelea stamountof 102

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procedure randLocalSearch ( region ; ubJammers ) 1 X ubJammers 2 while stoppingCriteria = FALSE do 3 randScatter ( region ;X ) 4 localOpt = FALSE 5 while localOpt = FALSE do 6 P leastJammedPoints ( region ) 7 moveJammers ( P ) 8 end 9 if allJammed = TRUE then 10 X X 1 11 end 12 end 13 return X endprocedure randLocalSearch Figure5-9:Pseudo-codefortherandomizedlocalsearchfor uncertainjamming. jammingenergyareassignedtotheset P .Then,thejammingdevicesaremovedalongagradient towardsthepointsin P untilthesearepointsarecovered.Severalmethodsareavai lablefor thefunction moveJammers includingthemethodofsteepestdescent[ 17 ]orthemoreefcient methodofconjugatedgradients[ 91 92 ].Theheuristicthendeterminesifallpointshavebeen jammed.Ifthisisthecase,theninline9wedecrementthenum berofjammingdevicesbyone andreturntoline2.Ifallpointsarenotjammed,werepeatth eloopuntileitherallpointsare coveredoruntilastoppingcriteriaismetinwhichcaseweex itthe while loop.Thenalvalueof X isreturnedasthesolutioninline13. AnexamplecanbeseeninFigure 5-10 .Forthisexample,apointrequires 3 unitsof jammingenergybeforeitisdeclaredtobejammed.Figure 5-10 (a)representstheplacement ofthejammingdevicesaccordingtotheoptimaluniformgrid solutionfromTheorem 25 .In thiscase, 400 devicesarerequired.InFigure 5-10 (b)weseetheassociatedcoveragefromthis solution.Thescallopshellaroundtheboundingboxcontain ingthenetworkindicatesthatinfact, theentireareaisjammed,butperhapsmoredevicesareusedt hanarenecessary.Insubgure (c),weseetheplacementofthe 298 jammingdevicesaccordingtotheheuristicsolution.Notic e 103

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-20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (a) (b) -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 25 -25 -20 -15 -10 -5 0 5 10 15 20 25 (c) (d) Figure5-10:Exampleofheuristicversusuniformplacement .(a)Deviceplacementonuniform grid.Thetotalnumberofjammingdevicesrequiredis400.(b )Coveragedisplayofuniform placement.(c)Heuristicjammerplacement.Thetotalnumbe rofrequireddevicesis298.(d) Heuristiccoverageplot.inFigure 5-10 (d)thatthecoverageoutsidetheboundingboxisreducedsig nicantlywhilestill jammingallpointsintheregion.Theheuristicreducesther equirednumberofdevicesby 25 : 5% Numericalresultsforseveralregionswithvariousrequire djamminglevelscanbeseenin Table 5-2 .Inthistable,welist,thesidelengthoftheregion( j a j ),therequiredjamminglevel ( L ),thenumberofjammingdevicesrequiredbytheupperandlow erboundscomputedusing Theorem 16 andTheorem 17 .Next,welisttherequiredjammingdevicescorrespondingt othe optimalgridstepsizewhichwasdeterminedusingabinaryse archwiththeupperandlower 104

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Table5-2:Numericalresultsareprovidedforseveralregio nswithvariousrequiredjamming levels.TheUpperBound,LowerBound,OptimalGrid,andLoca lSearchcolumnsprovidethe numberofjammingdevicesrequiredforthecorrespondingre gionaccordingtothetheorems presentedandtheproposedlocalsearch.ThePercentDecrea seshowsthesavingswhen comparingthelocalsearchtotheoptimalgridapproach. SideLevel Upper Lower Optimal Local Percent j a j L Bound Bound Grid Search Decrease 100 : 5 9 25 16 9 43 : 75 101 : 0 16 36 25 14 44 : 00 102 : 0 25 64 36 24 33 : 33 150 : 5 16 36 25 15 40 : 00 151 : 0 25 64 36 28 22 : 22 152 : 0 49 121 64 44 31 : 25 200 : 5 25 64 36 24 33 : 33 201 : 0 36 121 49 42 14 : 29 202 : 0 81 196 100 69 31 : 00 400 : 5 81 196 100 71 29 : 00 401 : 0 144 361 196 120 38 : 78 402 : 0 256 625 400 214 46 : 5 Total(Sum) 763 1909 1083 684 36 : 84 boundsastheinitialinterval.Finally,thenumberofjamme rsrequiredbythelocalsearchislisted alongwiththepercentdecreasefromthevaluecorrespondin gtotheoptimalgridsolution. 5.6ConcludingRemarks Inthischapter,weintroducedtheproblemofjammingacommu nicationnetworkunder completeuncertainty.Weexaminedthecasewhenthenetwork isknowntolieinasquarewith area a 2 .Wederivedupperandlowerboundsfortheoptimalnumberofj ammingdevicesrequired whentheyarelocatedattheverticesofauniformgrid.Weals oprovidedaconvergenceresult indicatingthattheproposedboundsaretight.Furthermore ,weprovedthatourapproachismore efcientthanthesolutionprovidedbyoptimallycoveringt hesquarewithcirclesofradius L Nextwepresentedarandomizedlocalsearchheuristicandco mparedthismethodtoplacingthe jammingdevicesonthenodesofauniformgridovertheareaof interest.Thelocalsearchwas abletoreducetherequirednumberofdevicesbyover 25% 105

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CHAPTER6 COOPERATIVECOMMUNICATION INMOBILEADHOCNETWORKS 6.1Introduction Inmanysituations,multiple“agents”worktogethertoachi eveasharedgoal.Cooperation betweentheagentsisimportanttoimprovetheefciencyand effectivenessbywhichtheir goalisreached.Ininformationsystemsthisideaalsoholds .Thatis,inwirelessnetworks, groupsofagentsareoftenemployedtoperformanumberofcoo perativetasksincludingthe synchronizationofinformationamongasetofusers,andthe accomplishmentofmissions inremoteareas.Insuchsituations,itisusefultomaintain collaborationamongtheagents performingthecooperativetasksinordertomaximizethepr obabilityofsuccess. Communicationisanimportantmeasureofcollaborationbet weenentitiesinvolvedina mission.Itallowsdifferentagentstoperformthesetoftas ksthathavebeenplanned,andatthe sametimetoimplementchangesinthecasethatanunexpected eventoccurs.Moreover,high communicationlevelsarenecessaryinordertoperformcomp licatedtasks,whereseveralagents mustbecoordinated.Wedescribeinthissectionthemaincon ceptsfoundintheliteraturerelated tooptimizingcommunicationtimeinadhocnetworksystems. Oneofthemaindifcultiesconcerningthemaintenanceofco mmunicationisanadhoc networkisdeterminingthelocationofagentsatagivenmome ntintime.Severalmethodshave beenproposedforimprovinglocalizationinthissituation .Mooreetal.[ 132 ],forexample, presentedalineartimealgorithmfordeterminingthelocat ionofnodesinanadhocnetworkin thepresenceofnoise.Otheralgorithmsforthesameproblem havebeensuggestedbyCapkunet al.[ 29 ],Dohertyetal.[ 60 ],andPriyanthaetal.[ 153 ]. Whilesuchalgorithmscanbeusefulindeterminingthecorre ctlocationofnodes,theyare onlyabletoprovideinformationaboutcurrentpositions,a ndarenotmeanttooptimizelocations foraspecicobjective.Packetrouting,ontheotherhand,h asbeenpreviouslystudiedwiththe goalofoptimizingsomecommonparameters,suchaslatency, costoftheresultingroute,and energyconsumed.Forexample,Butenkoetal.[ 25 ]proposedanewalgorithmforcomputinga 106

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backboneforwirelessnetworkswithminimumsize,basedona numberofrelatedalgorithmsfor thisproblem[ 23 33 129 ]. Anotherprobleminvolvingtheminimizationofanobjective functionoverallfeasible positionsofagentsinanadhocnetworkistheso-called locationerrorminimization problem.In theLOCATIONERRORMINIMIZATIONPROBLEM,givenasetofmeasurementsofnodepositions (takenfromdifferentsources),thegoalistodeterminease toflocationsforwirelessnodessuch thaterrorsinthegivenmeasurementsareminimized.Thispr oblemhasbeenformulatedand solvedusingmathematicalprogrammingtechniques,bytheu seofarelaxationforthegeneral problemintoasemi-deniteprogrammingmodel[ 19 20 46 165 ]. Therearemanyapplicationsofthisdescribedsystem.These includesituationswhere communicationinaregionisrequired,butnotopologically xedtransmissionsystemexists. Specicexamplesincludeemergency/rescueoperations,di sasterrelief,battleeldoperations, andBluetooth R r systems[ 137 ].Ineachoftheseexamplesthegoalsandobjectivesarexed inadvanceandcommunicationisimportantfortheattainmen tofthesegoals.Thecurrent technologiesusedinthesetypeofapplicationsallowimpro vedcommunicationsystemsthatrely onadhocwirelessprotocols.However,itisacombinatorial problemtodecidehowtomaintain communicationforthemaximumpossibletime,whenfacedwit htheinherentrestrictionsof wirelesssystems. Advancesinwirelesscommunicationandnetworkinghavelea dtothedevelopmentofnew networkorganizationsbasedonautonomoussystems.Amongt hemostimportantexampleof suchnetworkssystemsaremobileadhocnetworks(MANETs).M ANETsarecomposedofa setoflooselycoupledmobileagentswhichcommunicateusin gawirelessmediumviaashared radiochannel.Agentsinthenetworkactasbothclientsanda sserversandusevariousmulti-hop protocolstoroutemessagestootherusersinthesystem[ 137 141 ].Unliketraditionalcellular systems,mobileadhocnetworkshavenoxedtopology.Moreo ver,inaMANETthetopology changeseachtimeanagentchangesitslocation.Thus,theco mmunicationbetweentheagents dependsontheirphysicallocationandtheirparticularrad iodevices. 107

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InterestinMANETshassurgedintherecentyears,duetothei rnumerouscivilianand militaryapplications[ 155 ].MANETscanbesuccessfullyimplementedinsituationswhe re communicationisnecessary,butnoxedtelephonysystemex ists.Realapplicationsabound, especiallywhenconsideringadversarialenvironments,su chasthecoordinationofunmanned aerialvehicles(UAVs)andcombatsearchandrescuegroups. Otherexamplesincludethe coordinationofagentsinahostileenvironment,sensing,a ndmonitoring.Moregenerally, thestudyofprotocolsandalgorithmsforMANETsisofhighim portanceforthesuccessful deploymentofsensornetworks–whicharethemselvescompos edofalargenumberof autonomousprocessorsthatcancoordinatetoachievesomeh igherleveltasksuchassensing andmonitoring. ThelackofacentralauthorityinMANETsleadstoseveralpro blemsintheareasofrouting andqualityassurance[ 25 ].Manyoftheseproblemscanbeviewedascombinatorialinna ture, sincetheyinvolvendingsetsofdiscreteobjectssatisfyi ngsomedeniteproperty,suchasfor example,connectednessorminimumcost.Amongthechalleng ingproblemsencounteredin MANETs,wecanciterouting,orpathplanningasoneofthemos tdifculttosolve,becauseof thetemporarynatureofcommunicationlinksinsuchasystem .Infact,asnodesmovearound, theydynamicallydenetopologiesfortheentirenetwork.I nsuchanenvironment,itisdifcult todetermineiftwonodesareconnected,sinceanyoftheinte rmediatenodesmayleavethe networkatanytime. Thisscenariomakescleartheimportanceofclosecoordinat ionamonggroupsofnodes ifadenitegoalneedstobeattained.Ifatallpossible,apl anmustbedevisedsuchthat communicationamongnodesismaintainedforaslongaspossi ble.Withthisobjectivein mind,westudyinthischapter,aprobleminvolvingthecoord inationofwirelessusersinvolved inamissionoftasksthatrequireseachusertogofromaninit iallocationtoatargetlocation. Theproblemconsistsofmaximizingtheamountofconnectivi tyamongthesetofusers, subjecttoconstraintsonthemaximumdistancetraveledbyu sers,aswellasrestrictionson whattypesofmovementcanbeperformed.Theresultingprobl em,calledtheCOOPERATIVE108

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COMMUNICATIONPROBLEMONMOBILEAD-HOCNETWORKS(CCPMANET),isformally denedinalatersubsection. Thischapterisorganizedasfollows.Section 6.2 beginswithabriefreviewofsomeof thepreviousworkintheareasrelatedtocooperativecommun icationinwirelesssystems.Then wederivethediscreteversionoftheCCPMANET,referredtoasCCPMANET-D.Specically, weformulatetheproblemasacombinatorialproblemonagrap handprovideaninteger programmingformulation.InSection 6.3 ,weprovideasuiteofheuristicalgorithmsfor theproblembeginningwithasimpleconstructionalgorithm andculminatingwiththe implementationofaGRASPwithpath-relinkingfortheCCPMANET-D.Computationalresultsare presentedandthemethodscomparedinSubsection 6.3.6 .InSection 6.4 wederiveacontinuous formulationcalledCCPMANET-C.Acontinuousversionismorelikelytomodelreal-world scenariosaswenolongerrelyontheunderlyinggraphstruct ureandtheagentsarefreetomove subjecttokinematicconstraints.Finally,concludingrem arksandfutureresearchideasare presentedinSection 6.5 6.2DiscreteFormulations(CCPMANET-D) Asmentionedabove,adhocnetworksrepresentanextremelya ctiveareaofresearch[ 155 ]. Severalproblemsrelatedtorouting,powercontrol,andacc uratepositionupdate,havebeen studiedinthelastfewyears[ 136 ].Intermsofrouting,oneofthemainproblemsinad-hoc networksisthecomputationofanetworkbackbone.Theobjec tiveistondasubsetofnodes withasmallnumberofelementsthatcanbeusedtosendroutin ginformation.Theuseofsuch astructureisusefultosimplifythemanagementtasksrequi redbyaroutingprotocol.The backbonecomputationproblemcanbemodeledasaCONNECTEDDOMINATINGSET(CDS) problem.Here,theobjectiveistondasetofminimumsizefo rmingaconnectedbackbone,with theadditionalpropertythateachnetworkclientcandirect lyreachthisset.TheCDSproblem, whichcanbemodeledusingunitgraphs,hasseveralapproxim ationalgorithms[ 23 26 33 ],all ofwhicharebasedonapproximationpropertiesoftheMAXIMUMINDEPENDENTSETproblem onplanar,unitgraphs[ 12 ].Theuseofdiscreteoptimizationtechniquestomaximizec onnectivity 109

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inadhocsystemswasisarelativelynewideawhichwasputfor thbyOliveiraandPardalosin [ 137 ].WenowpresentsomediscreteformulationsfortheCCMPANET-D. Consideragraph G =( V;E ) ,where V = f v 1 ;v 2 ;:::;v n g representsthesetofcandidate positionsforthewirelessagents.Supposethatanodein G isconnectedonlytothosenodesthat canbereachedinoneunitoftime.Let U representthesetofagents, S = f s 1 ;s 2 ;:::;s j U j g V representthesetofinitialpositions,and D = f d 1 ;d 2 ;:::;d j U j g V thesetofdestinationnodes. Let N ( v ) 2 V ,for v 2 V ,representthesetofneighborsofnode v in G .Givenatimehorizon T ,theobjectiveoftheproblemistodetermineasetofroutesf ortheagentsin U ,suchthateach agent u i 2 U startsatasourcenode s i andnishesatthedestinationnode d i 2 D afteratmost T unitsoftime. Foreachagent u 2 U ,thefunction p t : U V returnsthepositionoftheagentattime t 2f 1 ; 2 ;:::;T g ,where T isthetimelimitbywhichtheagentsmustreachtheirdestina tions. Thenateachtimeinstant t ,anagent u 2 U caneitherremaininitscurrentlocation,i.e. p t 1 ( u ) ormovetoanodein N ( p t 1 ( u )) Wecanrepresentarouteforanagent u 2 U asapath P = f v 1 ;v 2 ;:::;v k g in G where v 1 = s u v k = d u ,and,for i 2f 2 ;:::;k g v i 2 N ( v i 1 ) [f v i g .Finally,if fP i g j U j i =1 istheset oftrajectoriesfortheagents,wearegivenacorresponding vector L suchthat L i isathreshold onthesizeofpath P i .Thisvalueistypicallydeterminedbyfuelorbatterylifec onstraintsonthe wirelessagents. Wenowhavetodecidewhattheactualmeasureofconnectivity amongsttheagentsontheir trajectorieswillbe.Obviously,thebestpossiblesituati onwouldbethecasewhenallagentsin thenetworkarelinked.However,inadhocsystems,thisisun likelyduetolimitsonpowerand fuel.Asnotedin[ 137 ],onepossiblemetricofconnectivityinagraphwouldbethe numberof connectedcomponentsinthegraphasaresultofthetrajecto riesoftheagents.Thismeasure hasamajordrawbackhowever,whichiseasilydemonstratedw iththefollowingexample. Consideragraphconsistingof n nodes.Ifthenumberofconnectedcomponentsis 2 ,thenwe areunsureastohowtointerpretthis.Itispossiblethatthe twocomponentscontain b n= 2 c and 110

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d n= 2 e nodesrespectively.Ontheotherhand,itisalsopossibleth atthetwocomponentscontain 1 and n 1 nodes.Thesearetwoverydifferentsituationsthatcanaris efromtheconnected componentsmetric,howeverwithanobjectivevalueofsimpl y 2 ,wehavenointuitionofthe resultingnetworkstructure. Anothermoreobjectivemeasureofconnectivityisasfollow s.Assumethattheagents haveomnidirectionalantennasandthattwoagentsinthenet workareconnectedifthedistance betweenthemislessthansomeradius r .Morespecically,let : V V 7! R represent theEuclideandistancebetweenapairofnodesinthegraph.T hen,wecandeneafunction c : V V 7!f 0 ; 1 g suchthat c ( p t ( u i ) ;p t ( u j )):= 8>><>>: 1 ; if ( p t ( u i ) ;p t ( u j )) r; 0 ; otherwise : (6–1) Withthis,wecandenetheCCPMANET-Dasthefollowingoptimizationproblem: max T X t =1 X u;v 2 U c ( p t ( u ) ;p t ( v )) (6–2) s.t. n i X j =2 ( v j 1 ;v j ) L i 8P i = f v 1 ;v 2 ;:::;v n i g ; (6–3) p 1 ( u )= s u 8 u 2 U; (6–4) p T ( u )= d u 8 u 2 U; (6–5) whereconstraint( 6–3 )ensuresthatthelengthofeachpath P i islessthanorequaltoits maximumallowedlength L i ThereaderisreferredtothepaperbyOliveiraandPardalos[ 137 ]foradditionalinteger programmingformulationsinwhichotherobjectivesarecon sideredanddiscussed.Wenishthis sectionbyprovidingtworesultsrelatedtothecomputation alcomplexityoftheproblem. Theorem19. FindinganoptimalsolutionforaninstanceoftheCOOPERATIVE COMMUNICATIONPROBLEMONMOBILEAD-HOCNETWORKSis NP -hard. 111

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Thisresult,duetoOliveiraandPardalos[ 137 ],followsbyareductionfromMAXIMUM3-SAT[ 79 ].Wenowextendthisresultinthefollowingtheorem. Theorem20. ConsideraninstanceoftheCCPMANET-D,with T asthetime-horizon.Findingan optimalsolutionateachtime-step t 2 [1 ;T ] is NP -hard. Proof. WewillshowthisresultbyreducingCLIQUEtoCCPMANET-Datanarbitrarytime-step. RecallthattheCLIQUEproblemisasfollows.Givenagraph G =( V;E ) andaninteger J j V j does G containaclique,orcompletesubgraph,ofsize J ormore[ 79 ]? ConsideraninstanceofCCPMANET-Datanytimestep t .Anoptimalsolutionisonein whichalltheagentsarepairwiseconnected.Thus,for n agentsthenumberofconnectionsinan optimalsolutionis n ( n 1) = 2 .Noticethatthisisequivalenttondingcliqueon n nodesofthe graph.Therefore,givenaninstanceofCLIQUE,byletting J = n ,wehavetheresult.Thusthere isabijectionbetweenoptimalcongurationsofagentsandc liquesinthegraph. Corrolary1. ForanyinstanceofCCPMANET-D,anupperboundontheoptimalsolutionisgiven by T u ( u 1) 2 ; (6–6) where T isthetimehorizonand u = j U j isthenumberofagents. Proof. ThisprooffollowsdirectlyfromTheorem 20 .Ifall u agentscommunicateatagiventime, thentheyformacliqueon u nodes.Thecliquewillcontain u ( u 1) = 2 verticesrepresentingthe communicationlinks.Iftheagentsmaintainthecliqueform ationoveralltimesteps,thenthe numberofcommunicationconnectionswillbe T u ( u 1) 2 (6–7) andthelemmaisproved. 112

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6.3AlgorithmsforCCPMANET-D 6.3.1ConstructionHeuristic Inthissection,weproposeaconstructionheuristictoquic klycreatehighqualitysolutions fortheCCPMANET-D.Ourobjectiveistoprovideafastwayofconstructingaseto fpaths, connectingwirelessagentsfromtheirinitialpositions S tothedestinations D suchthatthe resultingroutesarefeasiblefortheproblem.Theunionofs uchsequencesofnodeswilluniquely determinethecostofthesolution,whichiscalculatedusin gequation( 6–2 ).Thealgorithmalso triestocreatesolutionsthathaveaslargeavalueaspossib lefortheobjectivefunction. Thepseudo-codefortheconstructionheuristicisshowedin Figure 6-1 .Thealgorithmstarts initializingthecostofthesolutiontozero.Theincumbent solution,representedbythevariable solution ,isinitializedwiththeemptyset. Thenextstepconsistsofndingshortestpathsconnectinge achsource s i 2 S toa destination d i 2 D .Standardminimumcostowalgorithmscanbeusedtocalcula tethese shortestpaths.Forexample,theFloyd-Warshallalgorithm [ 77 170 ]canbeusedtocomputethe shortestpathbetweenallpairsofnodesinagraph.TheDijks traalgorithm[ 59 ]canalsobeused toperformthisstepofthealgorithm(withtheonlydifferen cethat,beingasingle-sourceshortest pathalgorithm,itmustberunfor j U j iterations,oneforeachofthe j U j source-destinationpairs). Intheloopfromlines 4 to 10 ,thealgorithmperformstheassignmentofnewpathstothe solution,usingtheshortestpathalgorithmdescribedabov e.First,asource-destinationpath s i d i isselected,andbasedonthisashortestpath P i correspondingtothispairisgenerated.Notice that,ifthelength(numberofedges)oftheshortestpath P i ismorethan T thereisnotfeasible solutionfortheproblem,sincethedestinationscannotber eachedattheendoftherequestedtime horizon.Thealgorithmchecksforthisconditiononline 6 Ifallsource-destinationpairsarefoundtobefeasible,th enasolutionisgeneratedbythe unionofall P i .Noticethatonceagent i reachesnode d i itcansimplyloiterat d i duringall remainingtime(untilinstant T ),asshowninline 7 .Thesequenceofnodesfoundasaresultof thisprocessisthenaddedtothesolutioninline 8 ofthealgorithm,andtheoptimumobjective 113

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procedure ShortestPath ( G;U;S;D;T ) 1 c 0 2 solution ; 3 Computeallshortestpaths SP ( s i ;d i ) foreachpair ( s i ;d i ) 2 S D 4 for i =1 to j U j do 5 P i SP( s i ;d i ) 6 if lengthof P i >T then 7 return ; 8 else 9 solution solution [P i 10 c c + newconnectionsgeneratedby P i 11 end 12 end 13 return ( c,solution ) endprocedure ShortestPath Figure6-1:Pseudo-codefortheshortest-pathconstructio nheuristic. valueisupdated(line 9 ).Finally,acompletesolutionisreturnedonline 11 ,alongwiththevalue ofthatsolution.Theorem21. Theconstructionalgorithmpresentedabovendsafeasible solutionfortheCCPMANET-Din O ( j V j 3 ) time. Proof. Afeasiblesolutionforthisproblemisgivenbyasequenceof positionsstartingat s i and endingat d i ,foreachagent u i 2 U .Clearly,theunionoftheshortestpathsprovidetherequir ed connectionbetweeneachsource-destinationpair,accordi ngtotheremarksinthepreceding paragraph;thereforethesolutionisfeasible.Supposetha t,inline 3 ,weusetheFloyd-Warshall algorithmforall-pairsshortestpath[ 77 170 ].Thisalgorithmrunsin O ( j V j 3 ) time.Then,at eachstepofthe for loopweneedonlytorefertothesolutioncalculatedbytheFl oyd-Warshall algorithmandaddittothevariable solution .Thiscanbedoneintime O ( j V j ) ,andtherefore the for loopwillruninatmost O ( j V jj U j ) time.Thusthestepwithhighesttimecomplexity istheoneappearinginline 3 ,whichimpliesthatthetotalcomplexityofthealgorithmis O ( j V j 3 ) 114

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6.3.2LocalImprovement Aconstructionalgorithmisagoodstartingpointintheproc essofsolvingacombinatorial optimizationproblem.However,duetothe NP -hardnatureoftheCCPMANET-D,suchan algorithmprovidesnoguaranteethatagoodsolutionwillbe found.Infact,itispossiblethatfor someinstancesthesolutionfoundbytheconstructionheuri sticisfarfromtheoptimum,andnot evenalocallyoptimalsolution. Toguaranteethatthesolutionfoundisatleastlocallyopti mal,weproposealocalsearch algorithmfortheCCPMANET-D.Alocalsearchalgorithmreceivesasinputafeasiblesolut ion and,givenaneighborhoodstructurefortheproblem,return sasolutionthatisguaranteedtobe optimalwithrespecttothatneighborhood. FortheCCPMANET-D,theneighborhoodstructureisdenedasfollows.Givenani nstance oftheCCPMANET-D,let S bethesetoffeasiblesolutionsforthatinstance.Then,if s 2S is feasiblefor ,theneighborhood N ( s ) of s isthesetofallsolutions s 0 2S thatdifferfrom s in exactlyoneroute.Obviously,consideringthisneighborho od,thereare j U j positionswhereanew pathcouldbeinserted;moreover,thenumberoffeasiblepat hsbetweenanysource-destination pairisexponential. Thus,inouralgorithm,insteadofexhaustivelysearchingt heentireneighborhoodforeach point,weprobeonly j U j neighborsateachiteration(oneforeachsource-destinati onpair).Also, becauseoftheexponentialsizeoftheneighborhood,welimi tthemaximumnumberofiterations performedtoaconstant MaxIter Weuserandomizationtoselectanewroute,givenasource-de stinationpair.Thisisdonein ourproposedimplementationusingamodiedversionofthed epth-rstsearchalgorithm[ 47 ]. A randomizeddepth-rst-search isidenticaltoadepth-rstsearchalgorithm,butateachst epthe nodeselectedtoexploreisuniformlychosenamongtheavail ablechildrenofthecurrentnode. Usingtherandomizeddepth-rstsearchweareabletondaro utethatmayimprovethesolution, whileavoidingbeingtrappedatalocaloptimumafteronlyaf ewiterations. 115

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procedure HillClimb ( solution ) 1 c f ( solution ) 2 while solution notlocallyoptimal and iter < MaxIter do 3 for i =1 to j U j do 4 solution solution nfP i g 5 P 0 i DFS ( s i ;d i ) 6 c' f ( solution [P 0 i ) 7 if lengthof P 0 i c then 8 c c' 9 iter 0 10 else 11 Restorepath P i 12 end 13 endfor 14 iter iter +1 15 endwhile 16 return ( solution ) endprocedure HillClimb Figure6-2:Pseudo-codefortheHillClimbingintensicati onprocedure. Adescriptionofthelocalsearchprocedureinformofpseudo -codeisgiveninFigure 6-2 Thealgorithmusedcanbedescribedasfollows.Initially,t healgorithmreceivesasinputthe basicfeasiblesolutiongeneratedonphase 1 (theconstructionphase).Aneighborhoodforthis solutionisthendenedtobethesetoffeasiblesolutionsth atdifferfromthecurrentsolutionby oneroute,aspreviouslydescribed. Giventhebasicfeasiblesolutionobtainedfromtheconstru ctionsubroutine,the neighborhoodisexploredinthefollowingmanner.Foreacha gent u i 2 U ,wereroutethe agentonanalternatefeasiblepathfrom s i to d i (lines 3 to 13 ).Recallthatapath P i isfeasible ifthetotallengthofthispathislessthan L i andtheagentreachesitstargetnodebytime T Thisalternatepathiscreatedonline5usingamodieddepth -rstsearchalgorithm[ 6 ].The modicationtotheDFSisarandomizationwhichselectsthec hildnodeuniformlyduringeach iteration.Thisprocedureisefcientinthatitcanbeimple mentedinpolynomialtime,asshown bellow. 116

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procedure OnePass ( G;U;S;D;T ) 1 solution ShortestPath ( G;U;S;D;T ) 2 solution HillClimb ( solution ) 3 return ( solution ) endprocedure OnePass Figure6-3:Pseudo-codefortheone-passheuristic. Theorem22. Thetimecomplexityofthealgorithmaboveis O ( kTu 2 m ) ,where k = MaxIter T isthetimehorizon, u = j U j and m = j E j Proof. Noticethatthemosttimeconsumingstepofthealgorithmist heconstructionofanew path(line 5 ).However,usingarandomizeddepth-rstsearchprocedure thiscanbedonein O ( m ) time[ 6 ].Eachiterationofthe while loop(lines 2 to 13 )willperformlocalimprovements inthesolutionusingthere-routingproceduretoimproveth eobjectivefunction.Anupperbound onthebestsolutionforaninstanceofthisproblemis Tu ( u 1) = 2 (thetimehorizonmultipliedby maximumnumberofconnections).Eachimprovementcanrequi reatmost MaxIter iterations tobeachieved.Therefore,intheworstcasethisheuristicw illendafter O ( kTu 2 m ) time. 6.3.3One-PassHeuristic ThetwoalgorithmsdescribedinSections 6.3.1 and 6.3.2 canbecombinedintoasingle one-passheuristicfortheCCPMANET-D[ 42 ].Thepseudo-codeforthecompletealgorithmused canbeseeninFigure 6-3 .Thenewalgorithmnowbehavesasasingle-start,diversic ationand intensicationheuristicfortheCCPMANET-D. Thetotaltimecomplexityofthisheuristiccanbedetermine dfromTheorems 21 and 22 Takingthemaximumofthetwotimecomplexitiesdeterminedp reviously,wehaveatotaltime of O (max f n 3 ;kTu 2 m g ) ,where T isthetimehorizon, u = j U j n = j V j m = j E j ,and k = MaxIter isthemaximumnumberofiterationsallowedonthelocalsear chphase. Thealgorithmproposedabovewastestedtoverifythequalit yofthesolutionsproduced,as wellastheefciencyoftheresultingmethod.Thetestinsta ncesemployedintheexperiments werecomposedof60randomunitgraphs,distributedintogro upsof20,eachgrouphaving graphswith50,75,and100nodes.Thecommunicationradiuso fthewirelessagentswasallowed 117

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Table6-1:Comparativeresultsbetweenshortestpathsolut ionsandheuristicsolutions. InstanceNodesRadius AgentsOnePassSPSoln AgentsOnePassSPSoln AgentsOnePassSPSoln 15020 1063.652.4 15152120.8 25414.66353.6 25030 1083.858.4 15182.2124.4 25516.2415.6 35040 1095.467.4 15228.6171.8 25695474.8 45050 10115.464.4 15275.8167.4 25797.4485.4 57520 1076.859 20270.2228.6 30575.2464 67530 1085.856 20299.6241.2 30725.4554 77540 1096.464.4 20386261 30862.6595.4 87550 1012567.8 20403.2246.8 301082.4670.8 910020 15113.6100.4 25333.4269.4 501523.21258.8 1010030 15166.2124.4 25511.2365 501901.41515.8 1110040 15203.4141 25600.6389.8 502539.21749.4 1210050 15255.8151.8 25756.8479.6 503271.22050.6 tovaryfrom20to50units.Thishasprovideduswithagreater baseforcomparison,resultingin randomgraphsandwirelessunitsthatmorecloselyresemble real-worldinstances. Thegraphsusedintheexperimentwerecreatedwiththealgor ithmproposedbyButenkoet. al[ 39 40 ]inthecontextoftheTDMAMESSAGESCHEDULINGPROBLEM.Theroutineswere codedinFORTRAN.RandomnumbersweregeneratedusingSchrage'salgorithm[ 166 ].Inall experiments,therandomnumbergeneratorwasstartedwitht heseedvalue 270001 Resultsobtainedinourpreliminaryexperimentsarereport edinTable 6-1 .Inthistable,the resultsoftheone-passalgorithm(OnePasscolumn)arecomp aredtoasimpleroutingscheme whereonlytheconstructionphaseisexplored(theSPSolnco lumn).Thesolutionsshowninthe tablerepresenttheaverageoftheobjectivefunctionvalue sfromthe 5 instancesineachclass. Thenumericalresultsprovidedinthetabledemonstratethe effectivenessoftheproposed heuristicwhentheimprovementphaseisaddedtotheprocedu re.Theproposedheuristic increasedtheobjectivevalueoftheshortestpathsolution sbyanaverageof 38% .Onereasonfor thisisthefactthat,whenagentsareroutedsolelyaccordin gtoashortestpath,theyarenottaking advantageoftheremainingtimetheyareallotted(i.e.thet imehorizon T )andthevaluesfrom thedistancelimitgivenby L 118

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procedure GRASP ( MaxIter ; RandomSeed ) 1 f 0 2 X ; 3 for i =1 to MaxIter do 4 X ConstructionSolution ( G;g;X ) 5 X LocalSearch ( X;N ( X )) 6 if f ( X ) f ( X ) then 7 X X 8 f f ( X ) 9 end 10 end 11 return X endprocedure GRASP Figure6-4:GRASPformaximization Ourheuristic,ontheotherhand,allowswirelessagentstot akefulladvantageofthese bounds.Thealgorithmcandothisbyadjustingthepathstoin cludethosenodeswithinthe rangeofotheragents.Inaddition,atanygiventimeanagent isallowedtoloiterinitscurrent position,possiblywaitingforotheragentstocomeintoits range.Thiscannotoccurinthe phase 1 algorithmbecause,accordingtotheshortestpathroutingp rotocol,loiteringisforbidden. Wenoticethatourmethodprovidessolutionsthatarebetter thantheshortestpath protocol.Thetimespentonthealgorithmhasalwaysbeenles sthanafewseconds,therefore thecomputationaltimeissmallenoughfortheproblemsizes exploredinourexperiments. Webelieve,however,thatthequalityofthesolutionsandco mputationaltimecanbefurther improvedusingabetterimplementation,andmoresophistic ateddatastructurestohandlethe informationstoredduringthealgorithm.6.3.4GreedyRandomizedAdaptiveSearch Inthissection,wedescribetheimplementationoftheGreed yRandomizedAdaptiveSearch Procedure(GRASP)(Section 2.7.2 )fortheCCPMANET-D.Pseudo-codeforthegenericGRASP isprovidedinFigure 6-4 .Wediscussinthissectionhowtheabovealgorithmcanbespe cialized toprovideapproximatesolutionsfortheCCPMANET-D.Inthefollowingsubsection,wedescribe 119

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analgorithmfortheGRASPconstructionphasethatprovides initialsolutionsforinstancesoftheCCPMANET-Dproblem.Thenweprovidealocalsearchalgorithmfortheimp rovementphase. ConstructionPhase .ThersttaskinaGRASPalgorithmistobuildgoodfeasibles olutions intermsofagivenobjectivefunction.Todothis,weneedtos pecifytheset A ,thegreedy function g ,theparameter ,andtheneighborhood N ( X ) ,for X 2 F .Thecomponentsof eachsolution X arefeasiblemovesofamemberofthead-hocnetworkfromanod e v toanode w 2 N ( v ) [f v g .Wesaythatforanagent u i 2 U locatedatnode v inthegraphthat P i ( v ) representsashortestpathfromthecurrentnode v tothedestinationforagent u i ,namelynode d i Thecompletesolutionisconstructedaccordingtothefollo wingprocedureoutlinedinthe pseudo-codereportedinFigure 6-5 .Inthegure, a h referstothecurrentlocationofanagent. First,thesolutionwhichisinitiallyemptyisaugmentedto includethestartinglocationsforall agents.Then,thetimevariable t isinitializedto1,andinline6anagent u i 2 U isselectedat randomandroutedfromitsalongashortestpath P i ( s i ) fromitssourcenode s i toitsdestination node d i .Ifthetotaldistanceof P i ( s i ) isgreaterthan L i ,thentheinstanceisclearlyinfeasibleand thealgorithmends.Otherwisetheprocedurecontinuesandt heremainingagentsarescheduled intheloopbeginningatline8.Theprocedureconsiderseach feasiblemove ( q;w;u ) before schedulinganagent.Afeasiblemoveconnectsthenalnode q ofasub-path P u ,for u 2 U ,to anothernode w ,suchthattheshortestpathfrom w to d u hasdistanceatmost L u P e 2 P u dist ( e ) Thesetofallfeasiblemovesinasolutionisdenedas A ( X ) Theloopfromlines12-14ensurethatanodecurrentlyatitsd estinationremainsthere. Likewise,theloopfromlines15-17scheduleanagent h onashortestpath P h ( a h ) fromits currentposition a h to d h ifthemaximumallowedtraveltimeforagent h isequaltothe jP h ( a h ) j From19-21,theset L A ( X ) isformedandconsistsofallfeasiblemovesforagentsnotye t scheduled.Theninline25,thegreedyfunction g returnsforeachmove k 2 L thenumberof additionalconnectionscreatedbythatmove.Asdescribeda bove,theconstructionprocedure willranktheelementsof L accordingto g .Thebest k elementsarethenaddedtotheRCLand 120

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procedure ConstructionSolution ( G;g;X ) 1 X ; 2 for i =0 to k do 3 X X [f s i g 4 end 5 t 1 6 selectrandomly u i 2 U androute u i onshortest-path P i ( s i ) 7 X X [f P i ( s i ) g 8 while t
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procedure LocalSearch ( X; MaxIterLS ) 1 X 0 ; 2 t 1 3 LastImprove 1 4 i 2 5 iter 0 6 while i 6 = LastImprove and iter < MaxIterLS do 7 Removecurrentpathfrom s i to d i foragent u i 8 while a i 6 = d i do 9 if dist ( a i ;d i )= L i P e 2 P i dist ( e ) then 10 Routeuser u i usingitsshortestpath P i ( a i ) 11 else 12 BestMove f ( a i ;l w ;u i ) j g (( a i ;l w ;u i )) >g (( a i ;l j ;u i )) ; 8 ( a i ;l j ;u i ) g 13 end 14 Add BestMove intothenewpathfor u i insolution X 0 15 t t +1 16 end 17 if f ( X 0 ) >f ( X ) then 18 X X 0 19 LastImprove i 20 end 21 i i +1mod k 22 iter iter +1 23 end 24 return ( X 0 ) endprocedure LocalSearch Figure6-6:LocalsearchforCCPMANET-D. objectiveistoimprovethesolutionasmuchaspossibleunti lalocaloptimalsolutionisfoundas describedinthepseudo-codeprovidedinFigure 6-6 Thelocalsearchreceivestheconstructionphasesolution X andaparameter MaxIterLS asinput.Ineachiteration,theneighborhood N ( X ) of X isexploredinsearchofasolution X 0 suchthat f ( X 0 ) >f ( X ) .Inordertoexplore N ( X ) ,aperturbationfunctionisdenedasfollows. Intheloopinlines5–21,agentsarere-routedusingagreedy methodsimilartothatofthe constructionphase.Inline6,thecurrentconstructionpha sepathforagent u i isremovedfromthe solution.Theneachfeasiblemoveisconsideredandthemove whichaddsthegreatestincrease totheobjectivefunction, BestMove ,isaddedtothenewpathforagent u i .Thisisrepeatedfor allagentsuntilanewfeasiblesolution X 0 2 N ( X ) A ( X ) iscreated.If f ( X 0 ) >f ( X ) ,then 122

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inline17, X 0 issetasthenewcurrentsolution.Theprocessreturnstolin e5andrepeatsuntilno agentcanbere-routedaccordingtothisgreedymethodandim provethecurrentsolutionoruntil somemaximumnumberofiterations MaxIterLS arecompleted. 6.3.5ComplexityoftheHeuristic Thefollowingtheoremsaddressthecomputationalcomplexi tyoftheproposedalgorithm. Theorem23. TheconstructionphasendsafeasiblesolutionfortheCCPMANET-Din O ( Tmu 2 ) time,where T isthetimehorizon, u = j U j ,and m = j E ( G ) j Proof. Noticethatthe while loopfromlines8-31willrequire T ( j U j 1) iterationsto complete.Likewise,theloopfromlines11-22requires j U j iterations.Withintheloop,the mosttimeconsumingstepistheconstructionofshortestpat h.However,thiscanbedoneusinga breadth-rstsearchin O ( m ) time[ 6 ].Thuswehavetheresult. Theorem24. Thetimecomplexityofthelocalsearchphaseis O ( kTu 2 m ) ,where T isthetime horizon, u = j U j m = j E ( G ) j ,and k = MaxIterLS Proof. TheproofissimilartoTheorem 23 .Noticethatthe while loopfromlines 5 22 perform localimprovementsaccordingthegreedyre-routingscheme .Againthemosttimeconsuming stepistheconstructionofashortestpathwhichcanbeaccom plishedin O ( m ) time.Each improvementcanrequireupto k iterationsoftheloop.Thuswehavetheproof. Corrolary2. TheoveralltimecomplexityoftheproposedGRASPis O ( lTu 2 m ( k +1)) ,where T isthetimehorizon, u = j U j m = j E ( G ) j k = MaxIterLS ,and l = MaxIter istheoverall numberofGRASPiterations.Proof. TheproofisimmediatefromTheorem 23 andTheorem 24 Path-relinking FirstintroducedbyGloverin[ 84 ],path-relinking(PR)wasusedasanenhancementfor tabusearchheuristics.PRwasrstcombinedwithGRASPbyLa gunaandMart[ 125 ].When appliedtoGRASP,path-relinkingintroducesamemorytothe heuristicwhichusuallyresults inimprovementsinsolutionquality.Thisisbecauseinthes tandardGRASPframework,the 123

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procedure PathRelinking ( x s ; E ) 1 x g randSelect ( y 2E :( x s ;y ) > ) 2 f max f f ( x s ) ;f ( x g ) g 3 x argmax f f ( x s ) ;f ( x g ) g 4 x x s 5 while ( x s ;x g ) 6 = ; do 6 m argmax f f ( x m ): m 2 ( x;x g ) g 7 ( x m ;x g ) ( x;x g ) nf m g 8 x x m 9 if f ( x ) >f then 10 f f ( x ) 11 x x 12 end 13 end 14 return x endprocedure PathRelinking Figure6-7:Path-relinkingsubroutine. multi-startnatureoftheheuristicdoesnotincludeanylon g-termmemorymechanismfor savingtraitsofgoodsolutionsgeneratedbythealgorithm. Path-relinkingallowsGRASPto rememberthesetraitsandfavortheminsuccessiveiteratio ns.GRASPwithpath-relinkinghas beensuccessfullyappliedtoproblemssuchasMAXIMUMCUT[ 71 ],QUADRATICASSIGNMENT[ 140 ],TDMAMESSAGESCHEDULING[ 40 ],andoriginallyforLINECROSSINGMINIMIZATION[ 125 ].ForasurveyofGRASPwithpath-relinking,thereaderisre ferredto[ 158 ]. Path-relinkingworksbymaintainingasetofelitesolution s E ,knownas guides and examinespoint-to-pointtrajectoriesbetweenaguidingso lutionandanincumbentsolution insearchofanoptimum.Pseudo-codeforagenericpath-reli nkingprocedureisprovided inFigure 6-7 .Toperformpath-relinking,webeginwithaguidingsolutio n x g 2E ,andan initialstartingsolution x s .Theguidingsolution x g isselectedatrandomfromthepoolofelite solutions E ,solongasthesymmetricdifference ( x s ;x g ) betweenthetwosolutions x s and x g issufcientlylarge.Thesymmetricdifferenceisdenedas thesetofpairwiseexchanges neededtotransform x s into x g .Recallthatallsolutionsin E arelocaloptima,andwearetrying todiscoversolutionswhicharenotlocatedintheneighborh oodsof x s or x g .Thereforethis 124

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constraintpreventsusfromapplyingpath-relinkingtosol utionswhicharetoosimilartoeach other,andwouldnotlikelyyieldanimprovedsolution[ 70 ]. Ateachstep,theprocedureexaminesallmoves m 2 ( x;x g ) ,andgreedilyselectsthemove whichresultsinthemaximumincreaseintheobjectiveofthe currentsolution.Thisoccursinline 6 ofthepseudo-codeinwhichthemove m isselectedasthemovewhichmaximizes f ( x m ) where x m isthesolutionwhichresultsfromincorporating m into x .Inline 7 ,thesymmetric differenceisupdated,andifnecessarythebestsolutionis updatedinlines 9 12 .Theprocedure endswhen ( x;x g )= ; ,i.e.when x = x g [ 158 ]. Path-relinkingcanbeappliedtoapureGRASPinastraightfo rwardmanner,whichcanbe visualizedinthepseudo-codeofFigure 6-8 .First,thesetofelitesolutions E isinitializedtothe emptysetinline 2 andisbuiltbyincludingthesolutionsfromtherst MaxElite iterations.After astandardGRASPiterationofgreedyrandomizedconstructi onandlocalsearchproducesalocal optimalsolution X ,the PathRelinking procedureiscalledonline 7 .FortheCCPMANET-D, theelementsinthesymmetricdifferencearetheagentpaths whichdifferbetweentheinitialand guidingsolutions.Thevalueof m fromFigure 6-7 isthepathforanagentinthesymmetric differencewhichresultsinthemaximumincreaseinthetota lnumberofcommunications betweentheagents.Inline 8 ,afunction UpdateElite iscalledinwhichtheelitepoolis possiblyupdated.Thesolutionreturnedfrompath-relinki ngisincludedintheelitepoolifit isbetterthanthebestsolutionin E orifitworsethanthebestbutbetterthantheworstandis sufcientlydifferentfromallelitesolutions[ 158 ].Finally,theoptimalsolutionisupdatedin lines 12 to 14 ifnecessary. 6.3.6ComputationalExperiments Theproposedprocedurewasimplementedinthe C programminglanguageandcompiled usingtheMicrosoft R r VisualC++6.0.Itwastestedona PC equippedwitha1800MHzIntel R r Pentium R r 4 processorand 256 megabytesofRAMoperatingundertheMicrosoft R r Windows R r 2000 Professionalenvironment. 125

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procedure GRASP + PR ( MaxIter ; RandomSeed ) 1 X ; 2 E ; 3 for i =1 to MaxIter do 4 X ConstructionSolution ( G;g;X ) 5 X LocalSearch ( X; MaxIterLS ) 6 if j E j = MaxElite then 7 X PathRelinking ( X; E ) 8 UpdateElite ( X; E ) 9 else 10 E E[f X g 11 end 12 if f ( X ) f ( X ) then 13 X X 14 end 15 end 16 return X endprocedure GRASP + PR Figure6-8:GRASPwithpath-relinkingformaximization. BoththepureGRASPandtheGRASPwithpath-relinkingwerete stedonasetof 60 randomunitgraphswithvaryingdensities 20 eachhaving 50 75 ,and 100 nodes.Theradiusof communicationvariesfrom 1 to 5 units(miles)inunitincrements.Wetestedeachcasewithth ree setsofmobileagentstoachievebettercomparisonsandmode lreal-worldscenarios.Thus,in total 900 testcaseswereexamined.Thegraphswerecreatedbyagenera torusedbyButenkoet al.[ 39 ]ontheTDMAMESSAGESCHEDULINGPROBLEM. SinceanyinstanceoftheCCPMANET-Discomposedofseveralparameters,i.e.thenumber ofmobileagents,theirrespectivesourceanddestinationn odes,theradiusofcommunication, andthemaximumtimehorizon,eachofwhichimpactstheoptim alsolutionfortheinstance,we willprovideournumericalresultsinseveralsetsoftables .First,wereportsolutionsforseveral representativeinstancesandprovideallinputparameters inordertoestablishaninferencebase fortheoverallexperiment.Thenwewillsummarizetheovera llresultsbyprovidingtheaverage solutionsforeachproblemset. 126

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Table6-2:Threeinstanceswithdifferentsetsofagentson5 0nodegraphsaregiven.Thevaluein the UBound columnwasfoundusingCorollary 1 Instance:50r30i1Nodes:50Agents:10MaxTime:10 Source:[6101035742106] Destination:[49474448464048424747] Radius GRASP GRASP+PR UBound 1 291 303 450 2 365 373 450 3 412 423 450 4 443 443 450 5 449 449 450 (a) Instance:50r30i1Nodes:50Agents:15MaxTime:10 Source:[1098968172555211] Destination:[49474448464048424747] Radius GRASP GRASP+PR UBound 1 756 757 1050 2 881 909 1050 3 963 972 1050 4 1029 1029 1050 5 1050 1050 1050 (b) Instance:50r40i4Nodes:50Agents:25MaxTime:10 Source: [89845446274217589381187568] Destination: [49484448464249404849454649454844424148434049454943 ] Radius GRASP GRASP+PR UBound 1 2613 2653 3000 2 2896 2918 3000 3 3000 3000 3000 4 3000 3000 3000 5 3000 3000 3000 (c) InTable 6-2 ,wereportsolutionsforthreedifferentinstanceson 50 nodegraphs.The Source vectorand Destination vectorprovidetherespective ( s i ;d i ) pairforeachagent respectively.Thespecicvaluesof s i wererandomlyselectedfromtherst 20% ofthenodes ofthegraph.Likewise,the d i valueswerechosenrandomlyfromthelast 20% ofnodes.This methodofselectionispreferredoveracompletelyrandomiz eddesignbecauseinreal-world situationssuchasacombatscenario,theavailableentryan dexitpointsfromabattlespaceare likelytobelimited.However,usingarandomselectionfrom theavailablesubsetofnodesallows formorethoroughtestingandhelpsavoidunintentionalbia ses. 127

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Table6-3:Threeinstanceswithdifferentsetsofagentson7 5nodegraphsaregiven.Thevaluein the UBound columnwasfoundusingCorollary 1 Instance:75r30i2Nodes:75Agents:10MaxTime:15 Source:[76133101315662] Destination:[68687173686873707462] Radius GRASP GRASP+PR UBound 1 571 575 675 2 614 621 675 3 658 658 675 4 670 670 675 5 675 675 675 (a) Instance:75r40i4Nodes:75Agents:20MaxTime:15 Source: [1157153894613143831014111593] Destination: [6366696162686267686662606166637372647171] Radius GRASP GRASP+PR UBound 1 2535 2554 2850 2 2746 2758 2850 3 2842 2842 2850 4 2850 2850 2850 5 2850 2850 2850 (b) Instance:75r30i1Nodes:75Agents:30MaxTime:15 Source: [141528104133451264923158512937711341154] Destination: [68696362657262626771656963646064606066647463736464 6365656063] Radius GRASP GRASP+PR UBound 1 4721 4870 6525 2 6002 6012 6525 3 6265 6285 6525 4 6497 6497 6525 5 6525 6525 6525 (c) Thecolumn MaxTime isthemaximumtimehorizon T .Recallthatallagentsmustreach theirdestinationnodebythistime.The GRASP columnprovidesthesolutionfromGRASPafter 1000 iterationsand UBound istheupperboundonthesolutionvalueandwascalculatedby the equationinCorollary 1 .Noticethatastheradiusvalueincreases,thenumberofcon nections betweentheagentstendstoconvergetothevalueoftheupper bound.Recallthattheupper boundvaluefromCorollary 1 isnotanupperboundontheoptimalsolutionforthegivengra ph perse;itisanupperboundonthesolutionforthegiventimeh orizonandnumberofagents. Thus,themoredensethegraph,thetighterthebound. 128

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Table6-4:A 100 nodeinstancewithsolutionswithradiusvaryingfrom 1 to 5 units.Thevaluein UBound wasfoundusingCorollary 1 Instance:100r30i2Nodes:100Agents:15MaxTime:20 Source:[9191018131812181586620181] Destination:[848883849696819583829380908581] Radius GRASP GRASP+PR UBound 1 1819 1821 2100 2 1960 1974 2100 3 2065 2067 2100 4 2100 2100 2100 5 2100 2100 2100 (a) Instance:100r30i1Nodes:100Agents:25MaxTime:20 Source: [1769199122157812863131617131317192521] Destination: [81898482889993899397849696919086988681898289818099 ] Radius GRASP GRASP+PR UBound 1 5183 5186 6000 2 5577 5647 6000 3 5898 5909 6000 4 5992 5992 6000 5 6000 6000 6000 (b) Instance:100r30i2Nodes:100Agents:35MaxTime:20 Source: [35112144451012141317417816715715131295619183161819 1052] Destination: [89958991849988919282988485898598928081859894828990 96919290969696998182] Radius GRASP GRASP+PR UBound 1 10222 10255 11900 2 11108 11224 11900 3 11660 11704 11900 4 11842 11845 11900 5 11900 11900 11900 (c) Table 6-3 presentsthespecicparametersandrelatedsolutionsfort hreeinstancesoftheCCPMANET-Don 75 nodegraphs.Onthesenetworks,thenumberofagentsvariedf rom 10 to 30 ,andthemaximumtimehorizonwas 15 .Again,weseethatasthecommunicationradius increasesthesolutionstendtotheupperboundvalues.Simi larresultsforthreegraphshaving 100 nodesareprovidedinTable 6-4 .Forthe 100 nodeinstances,thenumberofagentsvariedfrom 15 to 35 andthemaximumtraveltimewas 20 units.Theresultsfortheseinstancesalsoindicatethat theheuristicisrobustandabletoprovideexcellentsoluti onsforlargeinstances. 129

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Table6-5:AveragesolutionvaluesforGRASPandGRASPwithp ath-relinkingon50node graphs. Nodes Agents Radius GRASP GRASP+PR Bound 50 10 1 347 352.21 450 50 10 2 404.58 407.58 450 50 10 3 428.32 429.47 450 50 10 4 437.84 438.53 450 50 10 5 444.37 444.58 450 50 15 1 813.11 817.32 1050 50 15 2 937.74 945.47 1050 50 15 3 1001.11 1003.58 1050 50 15 4 1025.37 1026.21 1050 50 15 5 1037.16 1037.53 1050 50 25 1 2272.79 2315.58 3000 50 25 2 2686.26 2704.53 3000 50 25 3 2850.84 2861.95 3000 50 25 4 2924.05 2927.68 3000 50 25 5 2959 2959.26 3000 AverageCompTime(s) 2.89 4.29 – Table6-6:ComparativesolutionsofGRASPandGRASPwithpat h-relinkingon75nodegraphs. Nodes Agents Radius GRASP GRASP+PR Bound 75 10 1 574.95 577.42 675 75 10 2 629.42 631.37 675 75 10 3 653.53 654.63 675 75 10 4 665.42 665.89 675 75 10 5 669.47 669.84 675 75 20 1 2288 2319.63 2850 75 20 2 2639.37 2651.5 2850 75 20 3 2756.69 2762 2850 75 20 4 2805.53 2807.68 2850 75 20 5 2827.42 2828.42 2850 75 30 1 5349.84 5391.26 6525 75 30 2 6037.47 6064 6525 75 30 3 6310.90 6332.37 6525 75 30 4 6422.11 6430.80 6525 75 30 5 6472.42 6478.84 6525 AverageCompTime(s) 6.16 7.43 – Tables 6-5 6-6 ,and 6-7 showtheevolutionoftheaveragesolutionvaluesasthe communicationrangeincreasesforthe 50 75 ,and 100 nodegraphs,respectively.Notice oncemorethatasthecommunicationrangeincreases,theave ragesolutionconvergestothevalue 130

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Table6-7:ResultsofGRASPandGRASPwithpath-relinkingon 100nodegraphs. Nodes Agents Radius GRASP GRASP+PR Bound 100 15 1 1838.25 1840.45 2100 100 15 2 1996.75 2003.15 2100 100 15 3 2061.9 2064.7 2100 100 15 4 2083.1 2084.4 2100 100 15 5 2093.95 2094.05 2100 100 25 1 4979.1 5019.2 6000 100 25 2 5655.3 5674.35 6000 100 25 3 5869.35 5876.9 6000 100 25 4 5940.65 5944.7 6000 100 25 5 5978.2 5979.2 6000 100 35 1 9947.45 9997.15 11900 100 35 2 11254.55 11280 11900 100 35 3 11636.85 11664.5 11900 100 35 4 11787.9 11793 11900 100 35 5 11859.1 11860.35 11900 AverageCompTime(s) 5.17 8.05 – oftheupperboundgivenbyCorollary 1 .Inthesetableswealsoreporttheaveragecomputing timerequiredbyboththepureGRASPandtheGRASP+PRtondth eirbestsolutionswithin thespeciednumberofiterations.Foralloftheexperiment s,theGRASP+PRfoundsolutionsat leastasgoodasthepureGRASP,ndingsuperiorsolutionsfo r 45% oftheinstancestested. InFigures 6-9 6-10 ,and 6-11 ,weprovideplotsoftheaverageobjectivefunctionvalue versuscommunicationrangefoundusingGRASPwithpath-rel inking.Theupperboundvalues foreachcaseascomputedbyCorollary 1 arealsoplottedinthecharts.Thesegraphsindicatethat onaverage,astheradiusofcommunicationincreases,theob jectivefunctionvaluestendtothe upperboundvalues. 6.4AContinuousFormulation(CCPMANET-C) Inthissection,wepresentacontinuousformulationfortheCCPMANET-D.Thisformulation willprovideamorerealisticscenariothanthediscretefor mulationprovidedabove.Wewill assumethattheagentsareoperatinginbattlespace Q R d ,where Q isacompact,convexset withunitvolumeandtheEuclideannorm jjjj 2 in R d .Forourpurposes,wearegoingtoconsider theplanarcase,i.e. d =2 ,withtheunderstandingthatextensionstohigherdimensio nsare 131

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500 1000 1500 2000 2500 3000 1 2 3 4 5 ConnectionsCommunication radius 25 agents15 agents10 agents upper bound GRASP+PR upper bound GRASP+PR upper bound GRASP+PR 500 1000 1500 2000 2500 3000 1 2 3 4 5 ConnectionsCommunication radius 25 agents15 agents10 agents upper bound GRASP+PR upper bound GRASP+PR upper bound GRASP+PR Figure6-9:EvolutionofGRASP+PRsolutionvalueson50node graphsasthecommunication radiusincreasesfrom1to5units.possible.Supposethereare N wirelessagentsintheadhocnetwork.The N agentsareassumed tobeomnidirectionalandaremodeledaspointmasses.Wewil lsupposethattheagentsarefree tomovewithin Q atsomeboundedvelocity.Assumewithoutthelossofgeneral ity,thatthe maximumvelocitymagnitudeisunitary,i.e. jj ~v i ( t ) jj 1 ; for i 2f 1 ;:::;N g : Inordertoderiveacontinuousformulation,weneedtotode neanobjectivefunctionthat isconsistentwiththatofthediscreteformulation.Let R ij bethecommunicationconstantfor agents i and j .Thatis, R ij istheradiusofcommunicationforthetwoagents.Thenonepo ssible objectiveisaso-called heavysidefunction denedas H 1 R ij jj ~x itr ~x jtr jj 2 = 8>><>>: 1 ; if jj ~x itr ~x jtr jj 2 R ij ; 0 ; if jj ~x itr ~x jtr jj 2 R ij : (6–8) Agraphicalrepresentationof H 1 isgiveninFigure 6-12 .Whilethisfunctionwillworkasan objective,itisveryextreme.Thatis,if H 1 [ R ij jj ~x itr ~x jtr jj 2 ]=0 ,thenthereisnoinformation 132

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1000 2000 3000 4000 5000 6000 7000 1 2 3 4 5 ConnectionsCommunication radius 30 agents20 agents10 agents upper bound GRASP+PR upper bound GRASP+PR upper bound GRASP+PR Figure6-10:EvolutionofGRASP+PRsolutionvalueson75nod egraphsasthecommunication radiusincreasesfrom1to5units.providedwhichmightindicatewhereabettersolutionmight lie.Amoredesirablefunction wouldbeonethatapproximates H 1 butiscontinuous. Weconsidertoalternativesto H 1 .Therstfunctionisapiecewisecontinuousfunction givenas H 2 R ij jj ~x itr ~x jtr jj 2 = 8>>>>>><>>>>>>: 1 ; if jj ~x itr ~x jtr jj 2 R ij ; jj ~x itr ~x jtr jj 2 R ij +2 ; if R ij < jj ~x itr ~x jtr jj 2 2 R ij 0 ; if jj ~x itr ~x jtr jj 2 2 R ij : (6–9) ThisfunctionwhosegraphisprovidedinFigure 6-13 hasavalueequaltooneifagents i and j arewithinthecommunicationradius R ij ofoneanother.Thefunctionthendecreasesconstantly untiltheagentsare 2 R ij apartatwhichtimetheyareunabletocommunicate. 133

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2000 4000 6000 8000 10000 12000 1 2 3 4 5 ConnectionsCommunication radius 35 agents25 agents15 agents upper bound GRASP+PR upper bound GRASP+PR upper bound GRASP+PR Figure6-11:EvolutionofGRASP+PRsolutionvalueson100no degraphsasthecommunication radiusincreasesfrom1to5units. Thethirdandnalobjectivefunctionwewillconsiderisaco ntinuouslydifferentiable decreasingfunctionofthedistancebetweenagents i and j andthecommunicationradius R ij Thisfunction,givenas H 3 jj ~x itr ~x jtr jj 2 ;R ij = e jj ~x itr ~x jtr jj 2 R ij 2 ; (6–10) canbeseeninFigure 6-14 .Thisisperhapsthebestapproximationof H 1 inthatitcanbe interpretedastheprobabilityofagents i and j communicatingwhentheyaresomedistance jj ~x itr ~x jtr jj 2 apart. Nowthatwehavefoundasuitableobjectivefunctionwecande netheremaining parametersandconstraintsoftheproblem.Let ~x i ( t ) bethepositionofagent i attime t .Then,wecandescribethelocationofall N agentsatatime t 2 Z + asan N -vector X ( t )=( x 1 ( t ) ;x 2 ( t ) ;:::;x N ( t )) 2 Q N .Similarly,let ~v i ( t ) bethevelocityofagent i at time t .Therelationshipbetweenvelocityandpositionistypical andisgivenby ~v i ( t )= dx i dt 134

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Figure6-12:Theheavysidefunction H 1 InordertoformulatethecontinuoustimeanalogoftheCCPMANET-D,wemustconstrainthe maximumvelocityofeachagent.Thiswillenforcetheconstr aintsonthemaximumdistance traveledinthediscreteformulation.If S i 2 R 2 isthestartingpositionofagent i ,and D i 2 R 2 isthedestinationpointofagent i ,thenwecanformulatetheCONTINUOUSCOOPERATIVE COMMUNICATIONPROBLEMONMOBILEADHOCNETWORKS(CCPMANET-C)asfollows. Maximize Z T 0 X i
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Figure6-13:Alternateobjectivefunction H 2 Usingthisformulationasastartingpoint,heuristicsforc ontinuousglobaloptimization problemscanbeimplemented.Currentlywearedevelopingan algorithmbasedonthe ContinuousGreedyRandomizedAdaptiveSearchProcedure(C -GRASP)proposedbyHirsch etal.[ 101 ].SubsequentworkwithC-GRASPincludingenhancementsand stoppingrulescan befoundin[ 100 ].C-GRASPhasalsobeenusedtosolvesystemsofnonlineareq uations[ 99 ], andforsolvingcontinuousformulationsofdiscreteoptimi zationproblems[ 38 ].Thisworkis currentlyinprocessandtheresultswillappearinapaperla terthisyear[ 41 ]. 6.5ConcludingRemarks Inthischapter,weintroducedtheCOOPERATIVECOMMUNICATIONPROBLEMONMOBILE ADHOCNETWORKS.Wepresentedbothdiscreteandcontinuousformulations,d iscussedthe computationalcomplexity,andpresentedseveralalgorith msforsolvingeachformulation. Furthermore,extensivecomputationalresultswerepresen tedwhichshowtheeffectivenessof theproposedalgorithms.Lastly,wederivedacontinuousfo rmulationoftheproblem.Using 136

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Figure6-14:Secondalternateobjectivefunction H 3 thisversion,severalheuristicsforcontinuousoptimizat ionproblemscanbeappliedtoachieve solutionwhichmorecloselymirrorreal-worldsituations. Thisisbecausewehaveremovedthe underlyinggraphstructureandthemotionoftheagentsisco nstrainedkinematically. Futureworkontheproblemofpath-planningforagroupofwir elessuserssuchasthe onepresentedhereshouldcouldfocusonamultiobjectivepr obleminwhichtheagentsnot onlymaximizethecommunicationtimebutalsomaximizethea mountofthebattlespace(i.e. thecommunicationgraph)thatiscovered.Thisformulation wouldbeparticularlyusefulin combatsearchandrescueoperationsandotherreconnaissan ceapplications.Intheformulation considered,itisstillpossiblethattheagentswillloiter atagivennodeuntilsuchtimepassesthat theymustproceedtotheirdestinations.Insteadofmerelyc irclingasinglenode,theobjectiveof maximumbattlespacecoveragewillstillallowtheagentsto maximizethecommunication,while visitingmanyareasoftheregion. 137

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Inthefollowingchapter,wetakeacloserlookattheactualm echanismsusedfor communicationintheMANETdescribedinthischapter.Inste adofthegeneralizedviewof agentconnectivityusedhere,weexamineaparticulartypeo ftransceiverwhichmaybeused bytheagents.Inparticular,weconsiderthetimedivisionm ultipleaccess(TDMA)styleradios. TDMAtransceiversarepopularbecausetheymakeefcientus eoftheavailablebandwidthby allowingfrequencyreuse.Notsurprisinglyhowever,ittur nsoutthatseveralproblemsmustbe mitigatedinordertoensureeffectivegroupcommunication 138

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CHAPTER7 THETDMAMESSAGESCHEDULINGPROBLEM 7.1Introduction TheMANETsuchastheonedescribedinthepreviouschapteris anexampleofaso-called wirelessmeshnetwork (WMN).WMNshavebecomeanimportantmeansofcommunication in recentyears.Inthesenetworks,asharedradiochannelisus edinconjunctionwithsomepacket switchingprotocoltoprovidehigh-speedcommunicationbe tweenmanyusers.Thestations inthenetworkactastransmittersandreceivers,andarethu scapableofutilizingamulti-hop transmissionprocedure.Theadvantageofthisisthatsever alstationscanbeusedasrelaysto forwardmessagestotheintendedrecipient,allowingforbe yondlineofsightcommunicationfor stationsthataregeographicallydisbursedandpotentiall ymobile[ 39 ]. Meshnetworkshaveincreasedinpopularityintherecentyea rsandthenumberof applicationsissteadilyincreasing[ 155 ].Asmentionedin[ 8 ],WMNsallowuserstointegrate variousnetworks,suchasWi-Fi,theinternetandcellulars ystems.WMNscanalsobeutilizedin amilitarysettinginwhichtacticaldatalinksnetworkvari ouscommunication,intelligence,and weaponsystemsallowingforstreamlinedcommunicationbet weenseveraldifferententities.Fora surveyofwirelessmeshnetworks,thereaderisreferredto[ 8 ]. InWMNs,thecriticalprobleminvolvesefcientlyutilizin gtheavailablebandwidthto providecollisionfreemessagetransmissions.Unfettered transmissionbythenetworkstations overthesharedchannelwillleadtomessagecollisions.The refore,somemediumaccesscontrol (MAC)schemeshouldbeemployedtoschedulemessagetransmi ssionssothatcollisionsare prevented.Thetimedivisionmultipleaccess(TDMA)protoc olisaMACschemeintroduced byKleinrockin1987whichwasshowntoprovidecollisionfre ebroadcastschedules[ 117 ].In aTDMAnetwork,timeisdividedintoframeswitheachframeco nsistingofanumberofunit lengthslotsinwhichthemessagesarescheduled.Stationss cheduledinthesameslotbroadcast simultaneously.Thus,thegoalistoscheduleasmanystatio nsaspossibleinthesameslotsolong astherearenomessagecollisions. 139

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WhenconsideringthemessageschedulingproblemonTDMAnet works,thereare twooptimizationproblemswhichmustbeaddressed[ 175 ].Therstinvolvesndingthe minimum framelength ,orthenumberofslotsrequiredtoscheduleallstationsatl eastonce.The secondproblemisthatofmaximizingthenumberofstationss cheduledwithineachslot,thus maximizingthethroughput.Inthischapter,weconsidertheMESSAGESCHEDULINGPROBLEM ONTDMANETWORKS(MSP-TDMA).Weprovideaintegerprogrammingformulationand provethattheproblemis NP -hard.Wethenderiveseveralheuristicsandcomparetheir performanceagainstotheralgorithmsfortheliterature.E xtensivecomputationalresultsindicate thesuperiorityofourmethodsagainstreal-worldinstance s. 7.2ProblemDescription ATDMAnetworkcanbeconvenientlydescribedasagraph G =( V;E ) wherethevertex set V representsthestationsandthesetofedges E representsthesetofcommunicationlinks betweenadjacentstations.Therearetwotypesofmessageco llisionswhichmustbeavoided whenschedulinginTDMAnetworks.Therst,calledadirectc ollisionoccursbetween one-hop neighboringstations ,orthosestations i;j 2 V suchthat ( i;j ) 2 E .One-hopneighborswhich broadcastduringthesameslotcauseadirectcollision.Fur ther,if ( i;j ) 62 E ,but ( i;k ) 2 E and ( j;k ) 2 E ,then i and j arecalled two-hopneighbors .Two-hopneighborstransmittinginthe sameslotcauseahiddencollision[ 39 ]. Assumethatthereare M slotsperframe.Further,assumethatpacketssentatthebeg inning ofeachtimeslotandarereceivedinthesameslotinwhichthe yaresent.Let x : M V f 0 ; 1 g ,beafunctionwhere x mn := 8>><>>: 1 ; ifstation n scheduledinslot m ; 0 ; otherwise. (7–1) Also,let c : E !f 0 ; 1 g return1if i and j areone-hopneighbors,i.e.,if ( i;j ) 2 E and i 6 = j Asmentionedabove,therearetwoproblemswhichhavetobeso lvedinordertoobtain optimalbroadcastschedulesusingtheTDMAprotocol.Ther stistheFRAMELENGTH140

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MINIMIZATIONPROBLEM(FLMP)andthesecondistheTHROUGHPUTMAXIMIZATION PROBLEM(TMP).Usingtheaforementioneddenitionsandassumptions,we cannowformulate theMESSAGESCHEDULINGPROBLEMONTDMANETWORKS(MSP-TDMA)asthefollowing multiobjectiveoptimizationproblem: Minimize M Maximize M X i =1 j V j X j =1 x ij subjectto M X m =1 x mn 1 ; 8 n 2 V; (7–2) c ij + x mi + x mj 2 ; 8 i;j 2 V;i 6 = j;m =1 ;:::;M; (7–3) c ik x mi + c kj x mj 1 ; 8 i;j;k 2 V;i 6 = j;j 6 = k;k 6 = i;m =1 ;:::;M; (7–4) x mn 2f 0 ; 1 g ; 8 n 2 V;m =1 ;:::;M; (7–5) M 2 Z + : (7–6) Theobjectiveprovidesaminimumframelengthwithmaximumb andwidthutilization, whileconstraint( 7–2 )ensuresthatallstationsbroadcastatleastonce.Constra ints( 7–3 )and ( 7–4 )preventdirectandhiddencollisions,respectively.Weno teherethatwillnotbeattempting tosolvethisproblembyusingthetypicalmultiobjectiveop timizationapproach,inwhichone combinesthemultipleobjectivesintoonescalarobjective whoseoptimalvalueisaPareto optimalsolutiontotheoriginalproblem.Insteadwewillde coupletheobjectivesandhandleeach independently.ThisisdonebecausefortheMSP-TDMA,framelengthminimizationusuallytakes precedenceovertheutilizationmaximizationproblem.Thi sistheusualmodusoperandiusedby otherheuristicsintheliterature[ 164 169 175 ]. SupposethatwerelaxtheMSP-TDMAandonlytheconsidertherstobjectivefunction. ThisisreferredtoastheFRAMELENGTHMINIMIZATIONPROBLEM(FLMP)andisgivenby thefollowingintegerprogram: min f M : ( 7–2 )-( 7–6 ) g .Clearlyanyfeasiblesolutiontothis problemisfeasibleforMSP-TDMA.Now,consideragraph G 0 =( V;E 0 ) where V follows 141

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fromtheoriginalcommunicationgraph G ,butwhoseedgesetisgivenby E 0 = E [f ( i;j ): i;j aretwo-hopneighbors g .Thenusingthisaugmentedgraph,wecanformulatethefollo wing theorem.Theorem25. TheFRAMELENGTHMINIMIZATIONPROBLEMon G =( V;E ) isequivalentto ndinganoptimalcoloringoftheverticesof G 0 ( V;E 0 ) Proof. Recallthatinorderforamessagescheduletobefeasible,al lstationsmustbroadcast atleastonceandnocollisionsoccur,eitherhiddenordirec t.Noticenowthat E 0 containsboth one-hopandtwo-hopneighbors,andinanyfeasiblesolution ,neitherofthesecantransmitinthe sameslot.Thus,thereisaone-to-onecorrelationbetweent imeslotsin G andvertexcolorsin G 0 Hence,aminimumcoloringoftheverticesof G 0 providestheminimumrequiredslotsneededfor acollisionfreebroadcastscheduleon G AfteronehassuccessfullysolvedtheFLMPyieldinganoptimalframelength M ,then theTHROUGHPUTMAXIMIZATIONPROBLEM(TMP)givenasfollows max f P M i =1 P j V j j =1 x ij : ( 7–2 )-( 7–6 ) g canbesolved,where M isreplacedby M in( 7–2 )-( 7–6 ).Itturnsoutthat boththeFLMPandtheTMPhavebeenshowntobe NP -hard[ 39 67 ].Thusitisunlikelythat apolynomialalgorithmexistsforndingtheoptimalbroadc astschedule[ 79 ].Itisinteresting tonotehowever,thatifweignoreconstraint( 7–4 )whichpreventstwo-hopneighborsfrom transmittingsimultaneously,thentheresultingproblemi sin P ,andapolynomialtimealgorithm isprovidedin[ 93 ]. 7.3ComputationalComplexity Thissectionpresentsthecomputationalcomplexityresult sfortheMSP-TDMA.Itwasrst notedthattheMSP-TDMAis NP -completebyWangandAnsariin[ 169 ].However,theirproof ofthe NP -completenessoftherecognitionversionoftheproblemwas incorrectduetosome faultyarguments.Namely,theyclaimedthattheGRAPHCOLORINGPROBLEMisequivalent totheMAXIMUMINDEPENDENTSETPROBLEMbasedonanincorrectassumptionthat,given anarbitrarygraph,anoptimalcoloringcanbefoundbyrecur sivelycomputingamaximum independentsetandremovingitfromthegraph.Thus,bycolo ringdifferentindependentsetsin 142

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Figure7-1:CounterexampletotheclaimofWang&Ansarithat optimalgraphcoloringcanbe foundbyrecursivelyndingamaximumindependentsetandre movingitfromthegraph. differentcolors,theyclaimthatthechromaticnumberofth egraphequalsthetotalnumberof independentsetscomputed. Figure 7-1 presentsacounterexampletothisstatement.Itiseasytose ethatthe independencenumberofthegraphinthisgureis3.Assuming thattherstmaximum independentsetfoundusingtheso-claimed“optimal”color ingalgorithmofWangandAnsari is f 4 ; 5 ; 6 g andconsequentlyremovingthissetfromthegraph,weobtain aclique(complete subgraph)ofthethreevertices f 1 ; 2 ; 3 g .Theindependencenumberoftheremaininggraph is1,soallthreeoftheremainingverticeshavetobecolored indifferentcolors.Thus,the Wang-Ansaricoloringalgorithmresultsina4-coloring.Ho wever,itiseasytoseethatthe chromaticnumberofthisgraphis3.Forexample,oneoptimal coloringisgivenbythefollowing partition: f 1 ; 5 ; 6 g ; f 2 ; 4 g ; f 3 g .Therefore,thecoloringobtainedusingtheWang-Ansariap proach isnotoptimal. NextweprovethattherecognitionversionoftheMSP-TDMAisinfact NP -complete.We considerthefollowingproblem:TDMAM ESSAGE S CHEDULING P ROBLEM INSTANCE:Aundirectedgraph G =( V;E ) andaninteger K QUESTION:Doesthereexistabroadcastschedulewithframel ength K ? Theorem26. The TDMAMESSAGESCHEDULINGPROBLEMis NP -complete. 143

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Figure7-2:Constructionofgraph G 0 from G Proof. Toshowthat K -MSPis NP -complete,weneedtoshowthat(1) K -MSP2NP ;(2) Some NP -completeproblemreducesto K -MSPinpolynomialtime.Supposethat n = j V j and m = j E j .Withoutthelossofgenerality,weassumethat G isconnected(ifitisnot,wecan considereachconnectedcomponentseparately). K -MSP2NP sinceagivenbroadcastschedulewithframelength k K canbeveriedfor validityin O ( n 3 ) time.Indeed,thevericationofvalidityconsistsofcheck ing,foreachvertex i 2 V ,thattheset L i ofalltimeslotsinwhichtheverticesfrom f i g S N ( i ) transmitaccording tothegivenscheduledoesnotcontainanyrepeatedelements .Thiscanbedoneusingthesorting oftimeslotnumbersin L i in O ( j L i j +1)log( j L i j +1) timeforvertex i ,thereforethetotalrun timewillbe O n X i =1 ( j L i j +1)log( n ) = O ( m + n )log( n ) = O ( n 3 ) : Wewillshowthatthegraph k -coloringproblemcanbereducedto K -MSPinpolynomial time.Recallthatthe k -coloringproblemis,given G =( V;E ) andaninteger k ,doesthereexist apropercoloringoftheverticesof G thatuses k colors?Thisisawell-known NP -complete problem[ 79 ]. 144

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Givenagraph G =( V;E ) ,wewillconstructthecorrespondinggraph G 0 =( V 0 ;E 0 ) where V 0 = V S E and E 0 = [ i; ( i;j )]:( i;j ) 2 E;i;j 2 V S ( e 1 ;e 2 ): e 1 ;e 2 2 E .An exampleofthisgraphisshowninFigure 7-2 .Obviously G 0 canbeconstructedinpolynomial time.Moreover, G hasaproper k -coloringifandonlyif G 0 allowsabroadcastschedulewith framelength k + m .Toseethis,notethatbytheconstructionofgraph G 0 ( v 1 ;v 2 ) 2 E ifand onlyif v 1 ;v 2 2 V are2-hopneighborsin G 0 .Also, V 0 n V formsacliquein G 0 ,andanyvertexin thiscliqueisa2-hopneighborofanyvertexin V since G isconnected.Thusnoothervertexcan transmitinthesametimeslotwithavertexfromtheclique,s oanybroadcastschedulein G 0 will require m timeslotsjustforverticesfromthecliquetotransmit. Theremainingverticesin V 0 ( i.e. ,verticesfrom V )cantransmitaccordingtoanyproper coloringin G ,wheredifferenttimeslotsintheresultingbroadcastsche dulewillcorrespondto differentcolorsinthecoloring.Therefore,thereisaoneto-onecorrespondencebetweenproper coloringsin G andfeasiblebroadcastschedulesin G 0 .Weseethatinfact k -coloringreducesin polynomialtimeto K -MSP,where K = k + m .Thus,theproofiscomplete. 7.4Heuristics Inthissection,weintroduceanddiscussseveralheuristic swhichhavebeenappliedtotheMSP-TDMAwithvaryingdegreesofsuccess.Thespecicalgorithmwhic harecomparedinclude: GreedyRandomizedAdaptiveSearchProcedure(GRASP)[ 39 ],GRASPwithPathRelinking [ 28 ],ReactiveGRASPwithPathRelinking[ 27 ],SequentialVertexColoring(SVC)[ 175 ],Mean FieldAnnealing(MFA)[ 169 ],aMixedNeural-GeneticAlgorithm(HNN-GA)[ 164 ],andwe presentanewcombinatorialalgorithmbyCommanderandPard alos[ 43 ]. 7.4.1CombinatorialAlgorithmforTDMAMessageScheduling Theinherentintractabilityoftheproblemmotivatesthene edforefcientheuristicsto quicklyprovidegoodsolutionsfornon-trivialinstances. Inthissection,wedescribeanew algorithmfortheMESSAGESCHEDULINGPROBLEMONTDMANETWORKS.Theheuristic isatwo-phaseiterativeprocedureforwhichpseudo-codeis providedinFigure 7-3 .First,we 145

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procedure ComAlgBSP ( G 0 ) 1 M j V j 2 X 0 3 for i =1 to MaxIter do 4 M SlotMinimization ( G 0 ;; SlotIter ) 5 if M M then 6 M M 7 X BurstMaximization ( G 0 ;M ;V ) 8 end 9 if X X then 10 X X 11 end 12 end 13 return ( M ;X ;V ) endprocedure ComAlgBSP Figure7-3:Pseudo-codeoftheproposedheuristicforMSP-TDMA. concentrateprimarilyontheframelengthminimizationpor tionoftheMSP-TDMAbyusinga greedyheuristicforgraphcoloringwhichcomputesnearopt imalsolutionsfortheFLMP.Since thissolutionwillonlyhaveeachstationtransmittingexac tlyonce,alocalimprovementmethod isthenappliedwhichattemptstomaximizethethroughputwi thinthederivedframelength.To increasetheefciencyoftheprocedure,the BurstMaximization procedureisonlyenteredif thecurrentframelength M isasleastassmallasthecurrentbestvalue M .Aftersomespecied numberofiterations,thealgorithmterminatesreturningt hebestoverallsolution,whichconsists oftheframelength M ,thetotalnumberofbursts X ,andthescheduleofslotassignments V FrameLengthMinimization .Fortherstphaseofthealgorithm,weapplyagreedy constructionheuristictodeterminethevaluefor M ,thenumberoftimeslotsrequiredforall stationstotransmit.AsaresultofTheorem 25 ,themethodisbasedontheconstructionphase oftheGreedyRandomizedAdaptiveSearchProcedure(GRASP) [ 157 ]forcoloringsparse graphsproposedbyLagunaandMartin[ 126 ].Thisparticularmethodwaschosenbecauseit isabletoquicklyprovideexcellentsolutionsfortheframe length.Thatbeingsaid,anyother coloringheuristicwouldworknefortheframelengthminim izationphase.Infact,in[ 175 ]a methodbasedonSequentialVertexColoringwasusedtodeter minethevalueof M .However, 146

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procedure SlotMinimization ( G 0 ;; SlotIter ) 1 M 0 2 V 0 V 3 while V 0 6 = ; do 4 M M +1 5 Ecount 1 6 for j =1 to SlotIter do 7 ^ V V 0 U ; S ; 8 while ^ V 6 = ; do 9 if U = ; then 10 RCL f (1 )100% stationsofmaxdegreein ^ V g 11 else 12 RCL f (1 )100% stationsofmaxdegreein ^ V \ U g 13 endif 14 s randSelect ( RCL ) 15 S S [f s g 16 N ( s ) f w j ( s;w ) 2 E 0 g 17 ^ V ^ V= f s g[ N ( s ) 18 U U [ N ( s ) 19 endwhile 20 E f ( u;v ) 2 E 0 j u;v 2 V 0 =S g 21 if j E j
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Ourimplementationoftheframelengthminimizationheuris ticisexactlyasdescribed in[ 126 ].Theproceduretakestheaugmentedgraph G 0 ,aproportionalparameter ,anda value SlotIter asinputandcreatesaninitialbroadcastscheduleoneslota tatime.Thevalue 2 [0 ; 1] determinestheamountofrandomness,orconverselygreedin essthattheprocedure uses. SlotIter isthenumberofcandidateschedulesforaparticularslotfr omwhichthebestis chosen. Initially,theframelength M isinitializedto 0 and V 0 ,thesetofunscheduledstations isinitializedto V .Theinitialscheduleiscreatedinthe while loopfromlines 3 27 .After incrementingtheframelength,the for loopfromlines 6 25 isentered.Inthisloop, SlotIter candidateschedulesarecreatedforthecurrentslot V M .Initially, ^ V ,thesetofadmissible unscheduledstationsisinitializedto V 0 and U ,thesetofinadmissiblescheduledverticesis initializedtotheemptyset. S ,thesetofstationsscheduledinthecurrentslot,isalsose tto ; Fromlines 9 11 aso-calledRestrictedCandidateList(RCL)isconstructed andcontainsthe (1 )100% admissiblestationsofmaximumdegree.Itisnowclearhowth eparticularvalue of controlstheamountofrandomnessthatisusedbythealgorit hm.Avalueof =0 would resultinasimplerandomsearch,while =1 wouldyieldapuregreedysearch[ 152 ].Afterthe constructionoftheRCL,anelement s 2 RCLischosenatrandomandscheduledinthecurrent slotinline 15 .Thesets ^ V and U areupdatedandtheloopcontinues.Aftertheslotcapacityi s maximal,theset E iscomputedwhichcontainsthesetofedgesremainingintheg raphinduced bytheyetunscheduledstations.If j E j islessthanthecurrentminimumvalue Ecount ,thenthe currentcandidateslotscheduleissavedin V M inline 22 .Inline 26 after SlotIter samples,the bestslotscheduleisremovedfromthegraphandthemainloop repeats[ 126 ].Finally,theframe length M andthenalslotschedule V = f V 1 ;V 2 ;:::;V M g arereturnedtothemainprocedure. TheresultofthisprocedureisafeasiblesolutionforMSP-TDMAinwhicheachstation isscheduledtobroadcastinexactlyoneslotduringthefram e.Thisfollowsdirectlyfromthe resultproveninTheorem 25 .Foradiscussionofthecomputationalcomplexityofthepro posed procedure,thereaderisreferredto[ 126 ]. 148

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procedure BurstMaximization ( G 0 ;M ;V ) 1 X j V j 2 for i =1 to M do 3 T V 4 while T 6 = ; do 5 T f v j v 62 V i and 8 s 2 V i ; ( v;s ) 62 E 0 g 6 s randSelect ( T ) 7 V i V i [f s g 8 X X +1 9 endwhile 10 endfor 11 return X;V = f V 1 ;V 2 ;:::;V M g endprocedure BurstMaximization Figure7-5:Throughputmaximizationpseudo-code. ThroughputMaximizationPhase .Thesecondphaseoftheproposedmethodattempts tomaximizethethroughputbeginningwiththefeasiblesolu tionfoundintheframelength minimizationphase.Clearly,thesolutionfromtherstpha sewillnotprovideanoptimal throughputingeneral,becauseeachstationwillonlybesch eduledtotransmitonceintheframe. Therefore,weusearandomizedlocalimprovementmethodtos cheduleeachstationasmany timesaspossibleintheframe. Pseudo-codeforthethroughputmaximizationheuristicisp rovidedinFigure 7-5 ,andthe methodproceedsasfollows.Sinceeachstationisonlysched uledonce,thetotalnumberof bursts, X issetto j V j .Themainloopfromlines 2 10 locallyoptimizeseachslotintheframe. First,thesetofstationswhichcantransmitalongwiththos estationsalreadyscheduledin thecurrentslot,namely T isinitializedto V T isthenupdatedandcontainsthosestations v whicharenotalreadyscheduledinthecurrentslotandareno tadjacenttoanystation s which isscheduledinthecurrentslot.Anelementof T isthenselectedrandomlyandaddedtothe currentslot.Inline8thetotalnumberofburstsisincremen ted,andthelooprepeats.Themethod proceedstothenextslotwhentherearenostationswhichcan transmitwiththosecurrently scheduled,i.e.,when T = ; .Themethodreturnsthetotalnumberofbursts, X andtheupdated broadcastschedule V inline11. 149

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7.4.2GRASP Recallthatasdescribedabove,theGreedyRandomizedAdapt iveSearchProcedure (GRASP)isatwo-phaseiterativemetaheuristicforcombina torialoptimization[ 69 72 157 ].In therstphase,referredtoastheconstructionphase,agree dyrandomizedinitialfeasiblesolution iscreated.Theninthesecondphase,theinitialsolutionis improvedbytheapplicationofalocal searchprocedure.ThesolutionwhichisbestoutofallGRASP iterationsisreturned.GRASP hasbeenappliedtomanycombinatorialproblemssuchasquad raticassignment[ 128 140 ],job shopscheduling[ 18 7 ],privatevirtualcircuitrouting[ 156 ],andsatisability[ 154 ].GRASP wassuccessfullyappliedtotheMSP-TDMAbyCommanderetal.in[ 39 ].Wedescribethe implementationbelow. ConstructionPhase .TheconstructionphasefortheGRASPconstructsasolution iterativelyfromapartialbroadcastschedulewhichisinit iallyempty.Thestationsarerst sortedindescendingorderofthenumberofone-hopandtwo-h opneighbors.Next,aso-called RestrictedCandidateList (RCL)iscreatedandconsistsofthosegreedilyselectedsta tionswhich maybroadcastsimultaneouslywiththestationspreviously assignedtothecurrentslot.Fromthis RCLastationisrandomlychosenandassignedinthecurrents lot.AnewRCLiscreatedand anotherstationisrandomlyselected.Thisprocesscontinu esuntiltherearenostationstoputin theRCL,atwhichtimetheslotnumberisincrementedandthep rocedureisrepeatedrecursively forthesubgraphinducedbythesetofallverticeswhosecorr espondingstationshavenotyetbeen assignedtoatimeslot. LocalSearch .Thelocalsearchphaseusedisaswap-basedprocedurewhich isadapted fromasimilarmethodforgraphcoloringimplementedbyLagu naandMartin[ 126 ].First,the twoslotswiththefewestnumberofscheduledtransmissions arecombinedandthetotalnumber ofslotsisnowgivenas k = m 1 ; where m istheframelengthoftheschedulecomputedinthe constructionphase.Denotethenewbroadcastscheduleas f s m 0 ;n ;m 0 =1 ;:::;k;n =1 ;:::;N g Now,letthefunction f ( s ):= P ki =1 E ( m 0i ) ,where E ( m 0i ) isthesetofcollisionsinslot m 0i f ( s ) isthenminimizedbytheapplicationofalocalsearchproced ureasfollows. 150

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Acollidingstationinthecombinedslotischosenrandomlya ndeveryattemptismadeto swapthisstationwithanotherfromtheremaining k 1 slots.Afteraswapismade, f ( s ) is re-evaluated.Iftheresultisbetter,thatisif f ( s ) hasalowervaluethanbeforetheswap,theswap iskeptandtheprocessrepeatedwiththeremainingcollidin gstations. Ifaftereveryattempttoswapacollidingstationtheresult isunimproved,anewcolliding stationischosenandtheswaproutineisattempted.Thiscon tinuesuntileitherasuccessful swapismadeorforsomespeciednumberofiterations.Ifaso lutionisimprovedsuchthat f ( s )=0 ,thentheframelengthhasbeensuccessfullydecreasedbyon eslot.Thevalueof k isthendecrementedandtheprocessisrepeatedbeginningwi ththecombinationofthetwo “smallest”slots.Iftheprocedureendswith f ( s ) > 0 ,thennoimprovedsolutionwasfound. 7.4.3SequentialVertexColoring In[ 175 ],Yeoetal.takeamulti-objectiveoptimizationapproacht osolvingtheMSP-TDMA. Theyimplementatwo-phaseheuristicbasedontheideaofseq uentialvertexcoloring(SVC).In therstphase,theyonlyconsidertheproblemofminimizing theframelength.Theninphase2, theframelengthisxedwiththesolutionfromphase1andthe utilizationwithintheframeis maximized. FrameLengthMinimization .Forthisphase,theframelengthminimizationintheMSP-TDMAisattackedbysolvingthegraphcoloringproblemintheaugm entedgraph.More specically,analgorithmbasedonthesequentialvertexor deringmethodisusedtosolvethis problem.Thisisdonebyrstorderingthestationsindescen dingorderofthenumberofone-hop andtwo-hopneighbors.Therstvertexiscoloredandthelis toftheother N 1 vertices arescanneddownward.Theremainingverticesarecoloredwi ththesmallestcolorwhichhas notalreadybeenassignedtoaone-hopneighboringstation. Theprocessiscontinueduntilall verticeshavebeenassignedacolor. UtilizationMaximization .Beginningwiththisinitialschedule,phase2attemptsto maximizethethroughputintheTDMAframe.Tomaximizetheut ilizationwithintheframe whoselengthwasdeterminedinphase1,anorderingmethodof thesequentialvertexcoloring 151

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algorithmisapplied.Thestationsarenoworderedinascend ingorderofthethenumberof one-hopandtwo-hopneighbors.Therstorderedstationist henassignedtoanyslotsinwhichit cansimultaneouslybroadcastwiththepreviouslyassigned stations.Thisprocessisrepeatedwith everystationintheorderedlist.7.4.4MeanFieldAnnealing In1997,WangandAnsari[ 169 ]proposedaheuristicfortheMSP-TDMAbasedonMean FieldAnnealing(MFA).Instatisticalmechanics,thephysi calprocessofannealingisusedto relaxasystemtothestateofminimalenergy.Thisisdonebyh eatingthesoliduntilitmeltsand thencoolingitslowlysothatateachtemperaturethepartic lesrandomlyarrangethemselvesuntil reachingthermalequilibrium. In[ 116 ],Kirkpatricketal.introducedamethodforcombinatorial problemsknownas simulatedannealing (Section 2.7.2 ).Basedonthetheoryofthephysicalprocess,simulated annealingwasshowntoasymptoticallyconvergetothegloba lminimumafterperforminga numberofso-calledtransitionsatdecreasingtemperature s. Thoughsimulatedannealingisguaranteedtoconvergetothe globaloptimalsolution,this processisquiteoftencomputationallyexpensive.Meanel dannealing,aheuristicwhichmimics theideaofmeaneldapproximationfromstatisticalphysic s[ 150 ]canbeemployedinstead. InMFA,thestochasticprocessinsimulatedannealingisrep lacedbyasetofdeterministic equations.ThoughMFAdoesnotguaranteeconvergencetoagl obaloptimalsolution,it canprovideanexcellentapproximationtoanoptimalsoluti onandismuchlessexpensive computationally.7.4.5MixedNeural-GeneticAlgorithm AsinthealgorithmpresentedbyYeoetal.in[ 175 ],Salcedo-Sanzetal.[ 164 ]introduced atwo-phaseheuristicbasedoncombiningbothHopeldneura lnetworks[ 104 ]andgenetic algorithmsasin[ 171 ].Aswiththevertexcoloringalgorithm,phaseoneofthemix ed neural-geneticalgorithmminimizesframelengthandphase twoattemptstomaximizethe utilizationwithintheslot. 152

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FrameLengthMinimization .Theframelengthminimizationproblempresentedin[ 164 ] isthesameasdescribedabove.Forthesolution,adiscretetimebinaryHopeldneuralnetwork (HNN)isused.Asdescribedin[ 164 ],theHNNcanberepresentedasagraphwhoseverticesare theneurons(stations)andwhoseedgesarethedirectcollis ions.Thegraphisthenmappedtothe schedulematrix S asdenedabove.Theneuronsareupdatedoneatatimeafterar andomized initializationuntilthesystemconverges. UtilizationMaximization .Inthisphase,ageneticalgorithmisusedtomaximizethe channelutilizationwithintheframelengththatwasdeterm inedinphaseone.AHNNisalso usedtoensurethatallconstraintsaresatised.Genetical gorithmsreceivetheirnamefroman explanationofthewaytheybehave.Notsurprisingly,theya rebasedonDarwin'sTheoryof NaturalSelection.Geneticalgorithmsstoreasetofsoluti onsandthenworktoreplacethese solutionswithbetteronesbasedoncertaintnesscriterio nrepresentedbytheobjectivefunction value. 7.5ComputationalResults Theproposedheuristicwascodedinthe C ++programminglanguageandcompiledusing Microsoft R r VisualC++6.0.ThetestmachinewasaPCequippedwitha1700M HzIntel R r Pentium R r Mprocessorand1GBofRAMoperatingundertheMicrosoft R r Windows R r XP environment.Theheuristicwastestedonthreeclassicalin stancesaswellasasetof60random unitdiskgraphs[ 35 ]withvaryingdensities,20graphseachhaving50,75,and10 0nodes.The graphsarethosewhichwereusedbyButenkoetal.inapriorMSP-TDMAstudy[ 39 40 ]. Wecomparedourresultstothosefoundbyseveralheuristics fromtheliterature,allofwhich weretestedonthesamePCdescribedabove.AsmentionedbyPi tsoulisandResende[ 152 ], theparticularvalueof usedinrandomizedgreedyheuristicsistypicallydetermin edeither empiricallyorchosenrandomlyduringeachiteration.Alte rnatively,inaReactiveGRASPthe valueof istunedautomaticallytofavorspecicvaluesthattendtop roducebettersolutions. Nevertheless,duringourtesting,wefoundthatavalueof =0 : 1 generallyproducedthe bestoverallsolutionsfortheinstancestested.Theotherp arameter, SlotIter ,wassetto 5 153

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Inaddition,wehaveimplementedtheintegerprogramming(I P)modelfortheTHROUGHPUT MAXIMIZATIONPROBLEMusingtheXpress-MP TM optimizationsuitefromDashOptimization [ 108 ].Xpress-MPcontainsanimplementationofthesimplexmeth od[ 98 ],andusesabranch andboundalgorithm[ 173 ]togetherwithadvancedcutting-planetechniques[ 107 139 ].Thusnot onlyareweabletocompareourheuristictothoseintheliter ature,butwecanalsoseehowthe heuristicscomparewiththeoptimalsolutions. Thoughndingtheoptimalframelengthis NP -hard,wecanusetheIPmodelfortheTMPtoconrmwhetheraframelengthisoptimalornot.Considera ninstanceofMSP-TDMAandlet M betheoptimalframelength.Thenifweset M = M 1 intheintegerprogrammingmodel fortheTMP,theresultingIPwillbenotyieldanyfeasibleintegersolu tions.Infact,thelinear programmingrelaxationcouldalsobeinfeasible;thusimpl yingtheparticularinstanceoftheTMPisalsoinfeasible.Theproposedheuristicwasrsttestedu singthreeexamplesrstintroducedby WangandAnsariin[ 169 ]whichhavesincebecomethedefactotestcasesforTDMAbroa dcast schedulingalgorithms.Theseexamplesincludenetworksof varyingdensitieswith15,30,and40 stations.ThegraphsofthenetworkscanbeseeninFigure 7-6 Table 7-1 providestheoptimalsolutionsforthethreeaforementione dnetworksaswellas theheuristicsolutionsfoundbyourcombinatorialalgorit hm(ComAlg),theGRASPfrom[ 39 ], theMixedNeural-GeneticAlgorithm(HNN-GA)proposedin[ 164 ],theMeanFieldAnnealing (MFA)methodfromWangandAnsari[ 169 ],andtheSequentialVertexColoring(SVC)heuristic from[ 175 ].Thesolutionsarereportedas ( X;M ) .Noticethattheproposedalgorithmfoundthe optimalsolutionforeachofthethreeinstances.Theaverag ecomputationtimerequiredforthese instancesbyourmethodwas1.375s.Theaveragetimerequire dbyXpress-MPtocomputethe optimalsolutionswas 3411 : 4 seconds,withthe 30 stationnetworktaking 10212 seconds.Next,in ordertotestthescalabilityofthenewmethodandevaluatei tsperformanceforgeneralnetworks, wetestedthealgorithmsonthe 60 randomgraphsfrom[ 39 ]. 154

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1 3 7 2 5 6 12 4 9 10 11 8 13 14 15 (a) 25 26 27 28 29 30 9 1 6 4 5 3 8 2 7 10 12 15 13 14 11 18 20 16 24 17 21 22 19 23 (b) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 (c) Figure7-6:BenchmarkTDMAtestcases.(a)15stationnetwork.(b)30sta tionnetwork.(c)40station network.155

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Table7-1:Comparisonofsolutionsforthebenchmarkinstan cesfromWangandAnsari. Stations OptimalSoln ComAlg GRASP HNN-GA MFA SVC 15 (20,8) (20,8) (20,8) (20,8) (18,8) (18,8) 30 (36,10) (36,10) (36,10) (35,10) (39,12) (37,11) 40 (69,8) (69,8) (65,8) (67,8) (71,9) (60,8) Thecomparativeresultsoftheproposedalgorithmagainstt hebestsolutionscomputedby Xpress-MPafter 3600 seconds,aswellastheaforementionedheuristics 1 onthe50station graphsfrom[ 39 ]aregiveninTable 7-2 .Therstcolumnrepresentstheinstancenamefollowed bythedensityofthegraph G 0 .Noticethatthesolutionsfromthenewmethod(ComAlg)are atleastasgoodasanyotherheuristicforalloftheseinstan ces.Specically,thenewmethod providesbettersolutionsfor 15 ofthe 20 instances.Theasteriskimpliesthatthereportedsolution isoptimal.Fortheseinstances,thenewalgorithmfoundopt imalsolutionsfor 40% ofthetest cases.Theaverageframeutilizationisalsoreportedatthe bottomofthetable.Theutilization providesameasureoftheefciencyofabroadcastschedulea ndiscomputedasfollows := X M j V j : (7–7) Weseethatforthe 50 stationnetworks,theproposedalgorithmhasanaveragecha nnelutilization thatis 10 : 96% greaterthantheotherheuristics.Theaverageoptimalityg apforthethroughput maximizationphasewas 1 : 921% .Theaveragecomputationtimeforouralgorithmonthese instanceswas 2 : 8 seconds. Thecomparativesolutionsforthe 75 stationnetworksaregiveninTable 7-3 .Noticethatour methodoutperformstheotherheuristicsintheliteratureo neveryinstance.Forthesenetworks, theproposedalgorithmhasanaveragechannelutilizationt hatis 8 : 66% greaterthantheother methods.Theheuristicrequiredonaverage 6 : 62 secondstondthetargetsolution,andaswith the 50 stationnetworks,optimalframelengthsareachievedforal linstances.Furthermore,the 1 TheMFAalgorithmof[ 169 ]wasnotavailabletotheauthorsfortesting. 156

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Table7-2:Comparisonofoptimalandheuristicsolutionsforgraphswi th j V j =50 stations.Anindicatesthatthesolutionisoptimal,whileayindicatesthesolutionisthebestfoundbyXpress-MPafter 3600 s.Solutionsarereportedas ( X;M ) InstanceDensity Xpress-MP ComAlgGRASPHNN-GASVC 50r20i60.1136 (146,10) (145,10)(143,10)(145,10)(111,10) 50r20i20.0824 (86,6) (86,6)(84,6)(86,6)(82,6) 50r20i30.1040 (85,6) (85,6)(83,6)(85,6)(60,7) 50r20i70.0872 (90,6) (89,6)(87,6)(89,6)(52,6) 50r20i50.0968 (107,7) (107,7)(107,7)(105,7)(64,8) 50r30i10.1728 (78,8) (76,8)(74,8)(75,8)(54,9) 50r30i20.2122 (77,9) (75,9)(81,10)(70,9)(73,10) 50r30i30.1960 (84,9) (84,9)(78,9)(78,9)(78,10) 50r30i40.2048 (74,8)y (71,8)(67,8)(67,8)(60,10) 50r30i50.2096 (82,9) (79,9)(76,9)(84,10)(89,11) 50r40i10.3048 (76,12) (74,12)(73,12)(71,12)(58,14) 50r40i20.3680 (83,14)y (80,14)(77,14)(77,14)(83,16) 50r40i30.3408 (76,12)y (76,12)(80,13)(77,13)(56,15) 50r40i40.3712 (81,15) (81,15)(80,15)(76,15)(81,17) 50r40i50.3208 (71,12) (70,12)(67,12)(65,12)(55,14) 50r50i10.4280 (72,17) (72,17)(71,17)(75,18)(61,19) 50r50i20.4640 (61,15)y (61,15)y(65,16)(68,17)(55,17) 50r50i30.4480 (66.15)y (66,15)y(64,15)(65,16)(56,17) 50r50i40.4376 (70,15) (70,15)(72,16)(72,16)(79,18) 50r50i50.4088 (55,14)y (55,14)y(58,15)(56,15)(61,18) AvgSoln0.2603 (80.1,10.95) (80.1,10.95)(79.35,11.2)(79.3,11.35)(68.4,12.6) AvgUtil– 0.1463 0.14630.14170.13970.1086 solutionsfromourmethodarealwayswithin 10% ofoptimalsolutions,withanaveragegapof 5 : 8% Finally,thesolutionsforthe 100 stationnetworksaregiveninTable 7-4 .Oncemore,the newalgorithmndssolutionswhicharesuperiortotheother heuristicsforeachinstance.The utilizationwasanaverageof 10 : 17% higherthantheotheralgorithms.Theaveragecomputation timewas 12 : 17 seconds,withreportedgapsoflessthan 10% ofthebestsolutionfoundby Xpress-MPafter 3600 seconds.NoticealsothatXpress-MPwasunabletocomputeas olution superiortotheproposedheuristicfor 100r50i3 and 100r50i6 .Eachoftheseinstanceswereran for 10000 secondsandXpress-MPwasunabletocomputeafeasiblesolut ionintheframelength achievedbyourmethod. 157

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Table7-3:Comparisonofoptimalsolverandheuristicsolutionsforth e 75 stationnetworks. InstanceDensity Xpress-MP ComAlgGRASPHNN-GASVC 75r20i10.0988 (145,8) (139,8)(135,8)(136,8)(161,10) 75r20i20.1038 (122,8)y (119,8)(113,8)(112,8)(79,10) 75r20i30.1159 (113,7)y (108,7)(139,9)(121,8)(150,10) 75r20i40.0946 (116,7) (114,7)(109,7)(111,7)(84,8) 75r20i50.0988 (145,8) (138,8)(131,8)(135,8)(161,10) 75r30i10.1927 (114,12) (110,12)(117,13)(117,13)(91,13) 75r30i20.1867 (110,11) (105,11)(109,12)(101,11)(94,12) 75r30i30.2190 (140,15) (133,15)(132,15)(132,15)(81,17) 75r30i40.2009 (142,13)y (133,13)(127,13)(128,13)(144,15) 75r30i50.1927 (119,12)y (111,12)(106,12)(108,12)(89,12) 75r40i10.3328 (105,17)y (103,17)(104,18)(113,19)(79,20) 75r40i20.2980 (108,16)y (106,16)(109,17)(115,18)(86,19) 75r40i30.3403 (112,19)y (109,19)(105,19)(103,19)(87,20) 75r40i40.3492 (126,20)y (118,20)(124,21)(119,21)(79,24) 75r40i50.3143 (104,16)y (97,16)(100,17)(109,18)(114,20) 75r50i10.4587 (110,23)y (106,23)(107,24)(108,25)(123,29) 75r50i20.4622 (102,23)y (97,23)(99,24)(104,25)(108,27) 75r50i30.4807 (106,24)y (102,24)(104,25)(111,27)(114,29) 75r50i40.4750 (121,26)y (115,26)(112,26)(110,26)(102,28) 75r50i50.5088 (106,25)y (104,25)(111,27)(107,27)(106,28) AvgSoln0.2686 (118.45,15.5) (113.25,15.5)(114.65,16.15)(115,16.4)(106.6,18.05) AvgUtil– 0.1019 0.09740.09470.09350.0787 Asmentionedabove,thestrategyofrstminimizingthefram elengthandthenattemptingto maximizethethroughputwithinthisframeisacommonapproa ch[ 39 164 175 ].Inparticular, in[ 175 ],theauthorsproposeaheuristicbasedonsequentialverte xcoloringtoprovidea feasibleframelength.Then,agreedyheuristicisusedtoma ximizethethroughput.Similarly, Salcedo-Sanzetal.proposeahybridheuristicwhichminimi zestheframelengthusinganeural networkandthenmaximizesthethroughputusingagenetical gorithm[ 164 ]. Thealgorithmproposedin[ 39 ]basedontheGRASPmetaheuristicdescribedabove,usesa slightlydifferentstrategytoprovideapproximatesoluti onsfortheMSP-TDMA.Inthisalgorithm, theconstructionphasecreatesasolutioniterativelyinas imilarmannertotheoneproposed inthischapter.Themajordifferenceisthattheconstructo rin[ 39 ]doesnotcontaintheslot candidateconstructionloop(Figure 7-4 ,lines 6 25 ).Insteadthenalstationassignmentsin eachslotaretakenastherstproducedduringthegreedyran domizedconstruction.Thisis equivalenttosetting SlotIter equalto 1 inthe SlotMinimization methodabove.Bysetting 158

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SlotIter > 0 ,theproposedmethodismorelikelytoproducebettersoluti onsduringtheframe lengthminimizationphase. ThelocalsearchusedintheGRASPin[ 39 ]notonlyattemptstomaximizethethroughput withintheframecreatedduringtheconstructionphase,but alsotriestoreducetheframelength further.ThismethodwasadaptedfromtheGRASPforcoloring sparsegraphsofLagunaand Martin[ 126 ].Herethetwoslotswiththefewestbroadcastsarecombined creatinganew (infeasible)schedulewithonelessslotthanconstruction solution.Thesetofstationswhichcause messagecollisionsasaresultoftheslotcombinationisdet ermined.Foreachstationcausing acollision,everyattemptismadetoswapthemwithanothers tationfromtheremainingslots. Iftheswapreducesthenumberofcollisionsitiskeptandthe remainingcollidingstationsare considered.Ifallcollisionsaresuccessfullyaverted,th eprocessrepeatswiththecombination oftwomoreslots.Thiscontrastswiththeproposedmethodin thataftertheframelengthis determinedbythemorepicky SlotMinimization method,itisxedforthecurrentiteration. Experimentalanalysisshowsthatouralgorithmissuperior totheotherheuristicsinthe literature.Forall63instancestested,themethodfoundso lutionsatleastasgoodasanyofthe otheralgorithmsfromtheliteratureforallofthenetworks ,outperformingthemon 56 cases. Also,weseethatattemptingtosolvelarge-scaleinstances optimallyisimpractical.However, ourheuristicrequiredonly 7 : 49 secondsonaveragetondsolutionsthatarewithin 4 : 18% ofthe averagebestsolutionfoundbythecommercialIPsolverin 3600 seconds. 7.6ConcludingRemarks Inthischapter,wedescribedandimplementedseveralheuri sticsfortheMESSAGE SCHEDULINGPROBLEMONTDMANETWORKS.Inaddition,wehaveimplementedanoptimal solverusingXpress-MP[ 108 ].TheMSP-TDMAisanimportantproblemthatoccursinwireless meshnetworksregardingefcientlyschedulingcollisionf reebroadcastsforthenetworkstations. TheobjectiveoftheMSP-TDMAistwo-fold.First,thenumberofslotsrequiredtoschedule all stationsisminimized.Thenthethroughputistobemaximize dbyschedulingasmanystationsas 159

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Table7-4:Comparisonofoptimalsolverandheuristicsolutionsforne tworkswith j V j =100 stations. InstanceDensity Xpress-MP ComAlgGRASPHNN-GASVC 100r20i10.1006 (204,11)y (184,11)(179,11)(181,11)(110,11) 100r20i20.1028 (168,9)y (159,9)(169,10)(149,9)(131,11) 100r20i30.1138 (208,12)y (187,12)(177,12)(182,12)(101,14) 100r20i40.0992 (172,9) (160,9)(171,10)(153,9)(161,10) 100r20i50.1028 (183,10)y (170,10)(183,11)(164,10)(150,12) 100r30i10.1942 (138,14)y (132,14)(148,16)(137,15)(119,16) 100r30i20.1956 (158,15)y (147,15)1(152,16)(135,15)(145,17) 100r30i30.2270 (168,16)y (157,16)(164,18)(159,17)(138,18) 100r30i40.2172 (156,16)y (143,16)(140,17)(137,16)(102,17) 100r30i50.2088 (139,14)y (130,14)(151,17)(132,16)(150,17) 100r40i10.3552 (103,21)y (121,21)(136,24)(141,25)(142,26) 100r40i20.3300 (157,23)y (145,23)(149,24)(145,24)(141,25) 100r40i30.3480 (192,28)y (183,28)(178,28)(171,28)(115,31) 100r40i40.3164 (168,24)y (156,24)(148,24)(143,24)(153,26) 100r40i50.3178 (151,22)y (141,22)(149,24)(147,24)(134,25) 100r50i10.4826 (130,29)y (127,29)(142,33)(146,34)(104,36) 100r50i20.4604 (142,32)y (138,32)(137,32)(145,34)(148,36) 100r50i30.5076 (109,35)y (148,34)(149,35)(147,36)(138,38) 100r50i40.4690 (129,30)y (125,30)(128,32)(133,33)(126,33) 100r50i60.5012 (145,32)y (129,30)(149,34)(150,35)(125,35) AvgSoln0.2825 (156,20.57) (149.1,19.95)(155.95,21.4)(150.3,21.35)(131.65,22.7 ) AvgUtil– 0.0758 0.07470.07290.07040.0580 possiblewiththedeterminedframelength.Thealgorithmex ploitsthecombinatorialstructureof theprobleminordertoquicklyndhigh-qualitysolutions. Theproposedcombinatorialalgorithm,whichismosteffect ive,isexecutedintwophases, eachhandlingoneoftheobjectivesintheIPformulation.Th erstphasendsaminimalfeasible framelength,andthesecondphasemaximizesthethroughput withinthisframe.Experimental resultsindicatethatourheuristicoutperformsseveralot hermethodsfromtheliterature.In addition,themethodisrobustinthatitisabletondgoodso lutionsforawidevarietyof instances.Theefciencyofthemethodisdemonstratedbyco mparingtheruntimestothat requiredbyacommercialintegerprogrammingsolver. 160

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` ` ` ` ` ` `slot station 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 B 2 B B 3 B B 4 B B B 5 B B B 6 B B B 7 B B B 8 B B B (a) (b) (c) Figure7-7:ExampleGRASPbroadcastschedulesforthenetwo rksgiveninFigure 7-6 :(a)15 stationnetwork,(b)30stationnetwork,(c)40stationnetw ork. 161

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CHAPTER8 CONCLUSION Throughoutthisdissertation,wefocusedonoptimizationp roblemsintelecommunication systems,withaparticularemphasisonwirelessadhocnetwo rksoperatinginamilitary environment.Weexaminedtwobroadclassesofproblems;nam ely,thatofensuring communicationonthenetwork,andconverselyatproblemsof denyingserviceonthenetwork. Theproblemspresentedallhavesimilartraits.Forexample ,theyareallmodeledasdiscrete optimizationproblemsongraphs.Furthermore,aswesawall theproblemsweexaminedwere NP -hard.Wepresentedanin-depthlookatthecomputationalco mplexityofeachproblemand examinedwaysofdesigningefcientalgorithmsforeach. Chapter 1 providedanintroductiontothedissertationwithabriefde scriptionofthe majorcontributionsandadescriptionoftheproblemstofol lowinthesubsequentchapters.In Chapter 2 ,wepresentedanintroductiontoglobaloptimizationwhich includedmanytheorems andbasicdenitionswhichwereappliedinthelaterchapter s.Thischapterprovidedafoundation fortheworktofollow. Thesubjectsofthenextthreechaptersstudiedmethodsofde nyingserviceon telecommunicationnetworks.Theseproblemshelptoidenti fyweaknessesandvulnerabilitiesin networks.InChapter 3 ,webeganwiththestudyofjammingwiredtelecommunication networks. Wepresentedtwoformulationsandanalyzedtheircomputati onalcomplexity.Nextweprovided heuristicswhichprovidedexcellentsolutionsforrealand randomlygenerateddatasetsina fractionofthetimerequiredbyacommercialsoftwarepacka ge. InChapter 4 ,thegeneralWIRELESSNETWORKJAMMINGPROBLEMwasintroduced.This problemisanextensionoftheCRITICALNODEPROBLEMstudiedinChapter 3 .Weexamined severalvariationsoftheproblemandprovidedintegerprog rammingformulationsforeach. Furthermore,weprovidedformulationswhichincludedperc entileconstraints.Thecasestudies presentedshowedthattheadditionofthepercentileriskme asuresprovidedexcellentsolutions withasignicantreductionincost.Heuristicalgorithmsw erealsopresentedandacomputational 162

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studywaspresented.Foreachproblempresentedinthischap ter,weassumedthatthereasa prioriknowledgeofthenetworktobejammed.Wesawthateven withthisseeminglygenerous assumptiontheproblemsremained NP -hard. TheworkinChapter 5 relaxedthisassumptionandconsideredtheproblemofjammi ng anetworkwhen no informationwasassumedotherthanthegeneralareaknownto containthe network.Weconsideredasubproblemofplacingthejammingd evicesonalatticeoverlayingthe region.Arigorousanalysisfollowedinwhichwederivedupp erandlowerboundsontheoptimal numberofjammingdevicesrequiredtosuppressthenetwork. Weshowedthatbyconsidering thecumulativeeffectofthejammingdevices,thatourresul twassuperiortotheclassicalmethod ofcoveringaregionintheplanewithuniformcircles.Furth ermore,aconvergenceresultwas providedshowingthattheboundsaretightwithinaconstant .Toconcludethischapterwe presentedarandomizedlocalsearchalgorithmwhichbeganw iththelowerboundvaluederived andattemptedtominimizethenumberofdevicesneedtocover theregion.Experimentalresults indicatedthattheheuristicwasabletoreducethenumberof jammingdevicesbyapproximately 25% TheCOOPERATIVECOMMUNICATIONPROBLEMONMOBILEADHOCNETWORK S(CCPMANET)wasthetopicofChapter 6 .Thisproblemisconcernedwithdeterminingthe routesforasetofmobileagentsinsuchamannerthatcommuni cationamongsttheagentsis maximized.Weexaminedseveralobjectivefunctionsandcom paredtheirrelativeadvantages andprovidedanintegerprogrammingformulation.Wethenpr ovidedacomputationalstudy andprovedthattheproblemis NP -hardviaareductionfromthe3-SATISFIABILITYproblem. Further,weprovedthatitis NP -hardtocomputeanoptimalsolutionateachdiscretetimest ep. Next,wederivedseveralheuristicsandprovidedanextensi vecomputationalanalysis. InChapter 7 ,wetookacloserlookattheparticularcommunicationdevic esusedby theagentsintheCCPMANETproblem.Inparticular,weexaminedtheTDMAMESSAGE SCHEDULINGPROBLEM.TDMAisatypeoftime-divisionmultiplexingwheremultipl eusers sharethesamefrequencychannelbydividingthesignalinto differenttimeslots.Theusersare 163

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thenscheduledtobroadcastinasetoftimeslotssuchthatth ereisnointerferencebyuserswhich broadcastinthesameslot.Webeganbyexaminingtherecogni tionversionoftheproblemand showedthatitis NP -complete.Wefollowedthisbydesigningseveralheuristic sandcomparing theireffectivenessagainstotherheuristicsfromthelite rature. Astelecommunicationsystemsevolveeversorapidly,there areasmanydirectionforfuture researchasonecanimagine.Attheconclusionofeachchapte rweindicatedseveralproblems andextensionswhichcouldfollowfromthespecicworkatha nd.Ofcourse,thequestfor efcientalgorithmswithbetterworstcasecomplexitywill alwayslieattheforefrontforall theproblemsconsidered.Also,thedevelopmentoftightupp erandlowerboundswillcertainly aidallfutureendeavorsforeachproblem.Itismyhopethatm yworkrepresentsthecurrent state-of-the-artfortheproblemspresented,andthatmyef fortswillhelpourmilitaryperform betterastheyfacethedauntingtaskofdefendingourfreedo mswherevertheyarecalled. 164

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BIOGRAPHICALSKETCH ClaytonW.CommanderwasborninFt.WaltonBeach,Florida,o nAugust 23 ; 1982 .He wasraisedinnearbyNiceville,Florida,andgraduatedfrom NicevilleHighSchoolin 2000 AfterreceivinganAssociateofArtsdegreefromOkaloosa-W altonCommunityCollege,Clayton enrolledintheDepartmentofMathematicsattheUniversity ofFloridainAugustof 2001 .In May 2003 ,hegraduateds umma c um l audeandbeganworkingfortheUnitedStatesAirForce. InJanuary 2004 ,whileworkingatEglinAirForceBase,Claytonenteredgrad uateschool intheDepartmentofIndustrialandSystemsEngineeringatt heUniversityofFloridaandbegan studyingoptimizationwithProfessorPanosPardalos.Theh appiestdayofClayton'slifecameon June 18 ; 2005 ,whenhemarriedtheloveofhislife,LeahSusi.Hereceivedam aster' sdegreein December 2005 andearnedhisPh.D.inAugust 2007 .ClaytonandLeahlivehappilywiththeir twochihuahuasReinaandIsabelleinNiceville,FL. 177