On Approximation of New Opimization Methods
for Assessing Network Vulnerability
Thang N. Dinh, Ying Xuan, My T. Thai
CISE Department
University of Florida
Gainesville, FL, 32601
Email: {tdinh, yxuan, mythai }@cise.ufl.edu
E.K. Park
CSEE Department
University of Missouri at Kansas City
Kansas City, MO 64110
Email: ekpark@umkc.edu
Taieb Znati
CS Department
University of Pittsburgh
Pittsburgh, PA 15215
Email: znati@cs.pitt.edu
AbstractAssessing network vulnerability before potential dis
ruptive events such as natural disasters or malicious attacks is
vital for network planning and risk management. It enables us to
seek and safeguard against most destructive scenarios in which
the overall network connectivity falls dramatically. In this paper,
we study a new problem called 3disruptor. The goal is to identify
nodes (links) in the network whose removal results in a large
number of disconnected pair of nodes i.e. there is no functional
path between them.
We formulate the problem as a graphbased optimization
problem and prove that detecting such a set of nodes (edges)
in general case is NPcomplete, thus it cannot be solved in
polynomial time unless P = NP. We present an 0 (log n log log n)
pseudoapproximation algorithm for detecting the set of nodes
and an O(log1.5 n) pseudoapproximation algorithm for detecting
the set of edges. Our proposed approximation algorithms can
handle both homogeneous and heterogeneous networks with
unidirectional links.
I. INTRODUCTION
Connectivity plays a vital role in network performance and
is fundamental to vulnerability assessment. Potential disruptive
events, such as natural disasters or malicious attacks, which
always destroy a set of interacting elements or connections,
can dramatically compromise the connectivity and result in
considerate decline of the network QoS, or even breakdown
the whole network [1] [2] [3] [4]. Of this concern, pre
active evaluation over the network vulnerability with respect
to connectivity, in order to defense such potential disruptions,
is quite essential and beneficial to the design and maintenance
of any infrastructure networks, for example, communication,
commercial, and social networks.
Most studies over network vulnerability abstract the network
as a graph G = (V, E), which consists of a set of vertices V
and a set of edges E representing the communication links.
Due to the inhomogeneity of general graphs, it is often the
case that removing some vertices and edges will decrease the
network connectivity to a greater extent than removing other
ones. Therefore, these vertices and edges are more critical
to the overall graph connectivity, hence the corresponding
elements and connections in the network reveal a higher risk
in the front of potential disruptions.
There have been numerous efforts on proposing evaluation
measures of the network vulnerability, as summarized in
[1]. On one hand, several global graph measures, such as
Cyclomatic number, Maximum network circuits, Alpha index,
and Beta index, which investigate basic graph properties,
i.e., number of vertices, edges and pairwise shortest paths,
are adopted to evaluate the network vulnerability. However,
these global measures can neither be rigorously mapped to
the over network connectivity, nor reveal the set of most
critical vertices and edges, thus are not suitable to assess
the network vulnerability in terms of connectivity. On the
other hand, researchers focused on local nodal centrality [5],
such as degree centrality, betweenness centrality and closeness
centrality, in order to differentiate the critical vertices from
the others, and further evaluate the network by quantifying
such vertices. Unfortunately, being unable to cast these local
properties to global network connectivity, these measures fail
to indicate accurate vulnerabilities and cannot reveal the global
damage done on the network under attacks.
Instead of such detours, we model the objective network
as a connected directed graph, and directly quantify the
minimized set of vertices/edges whose removal incurs a certain
level of network disruption, i.e., reduces the overall pairwise
connectivity to some certain value as a measure for the
objective graph, where the connectivity for each vertex pair is
quantified as 1 if they are strongly connected and 0 if not. The
motivation behind this is to explore the number of necessary
disruptive events to incur a certain level of disruption in the
objective network, i.e., the more vertices/edges required to
be removed, the less vulnerable the network is; conversely,
the fewer vertices/edges needed to removed, the easier this
network is to be destroyed.
In the light of the above arguments, the following new op
timization problem: finding a minimized set of vertices/edges
whose removal degrades the pairwise connectivity to a desired
degree has to be solved in order to assess the vulnerability
of the objective network. Considering that disrupting these
vertices and edges will considerately degrade the network
performance, we refer to them as 3disruptor throughout
this paper, where 0 < 3 < 1 denotes the fraction of
desired pairwise connectivity (which we will define later).
Two new optimization problems 3vertex disruptor and 3edge
disruptor will be studied and proved to be NPcomplete. We
addressed them with several pseudoapproximation algorithms
with a provable performance bound, which thus ensure the
(a) network A
(b) network B
Fig. 1. After the "central" vertex (in black) with maximum outgoing degree
is removed, A is still strongly connected while B is fragmented; however in
fact, only removing one vertex (in grey) is enough to destroy A.
feasibility and accuracy of this evaluation measure.
The benefit of our new measure can be briefly illustrated in
Fig.l, compared with the assessment using degree centrality.
Notice that both networks A and B are with 7 vertices and
originally strongly connected. According to the nodal degree
centrality, removing the black vertex with maximum outgoing
degree 5 in Fig.l(a) leaves the network A still strongly
connected with 5 vertices; and removing the black vertex
with maximum outgoing degree 4 in Fig.l(b) partitions the
graph into two strongly connected components. In this sense,
network A is somewhat stronger (less vulnerable) than B.
However, our model can discover that, deleting only the grey
vertex in A will be enough to bring the overall connectivity
to 0; on the contrary, at least 3 vertices in B are required
to be removed to make overall connectivity 0. Therefore, A
is actually much more vulnerable. Apparently, our measure
provides more accurate assessment.
Furthermore, our study over the multiple disruption levels
(different values of 3) presents a deeper meaning and greater
potentials. Several recent studies in the context of wireless net
works have aimed to discover the nodes/edges whose removal
disconnects the network, regardless of how disconnected it is
[6] [7] [8] [9]. Apparently, this is a weaker version of our 3
disruptor, since no specification over the quantified network
connectivity is concerned. However, it is not reasonable to
limit the possible disruption to only disconnecting the graph,
ignoring how fragmented it is. For instance, a scalefree
network can tolerate high random failure rates [10], since
the destruction to boundary vertices may not significantly
decline the network connectivity even though the whole graph
becomes disconnected. In addition, different disruption levels
may require different sets of disruptor on which our model can
differentiate whereas existing methods cannot. For example,
the node centrality method always returns a set of nodes with
nonincreasing degrees regardless of the disruption level.
The main contributions of this paper are as follows:
Introducing a novel quantitative function based on an op
timization problem /disruptor on general graphs, which
consists of two versions 3vertex disruptor and 3edge
disruptor;
Proving the NPcompleteness of the two problems above
and further proving that no PTAS exists for 3vertex
disruptor;
Providing an O(log2 n) pseudoapproximation algorithm
for /vertex disruptor, and an O(log n log log n) pseudo
approximation algorithm for 3edge disruptor. These
solutions can be applied to both homogeneous networks
and heterogeneous networks with unidirectional links and
nonuniform nodal properties.
The paper is organized as follows. We include the problem
definition, models and notations in Section II. We provide
the hardness results in Section III. The pseudoapproximation
algorithms for 3edge disruptor and /vertex disruptor are
presented in Section IV and Section V respectively. Related
work can be found in Section VI. Section VII summarizes the
whole paper.
II. MODEL AND DEFINITIONS
Besides the homogeneous network model consisting of uni
form nodes and bidirectional links, the heterogeneous network
model, where various interacting elements of different kinds
are connected through unidirectional links with nonuniform
expenses, finds numerous applications nowadays [11][13],
but as well, exhibits multiple difficulties for optimization and
maintenance. In the light of this, we abstract our general
network model as a directed graph G(V, E), where V refers to
a set of nodes and E refers to a set of unidirectional links. The
expense of each directed edge (u, v) between vertex u and v is
quantified as a nonnegative value c(u, v), for all the IE m
links among VI n nodes. As mentioned above, our
evaluation over the network vulnerability is based on the value
of overall pairwise connectivity in the abstracted graph, which
is defined as follows: given any vertex pair (u, v) E V x V
in the graph, we say that they are connected iff there exists
paths between u and v in both directions in G, i.e., strongly
connected to each other. The pairwise connectivity p(u, v) is
quantified as 1 if this pair is connected, 0 otherwise. Since the
main purpose of network lies in connecting all the interacting
elements, we study on the aggregate pairwise connectivity
between all pairs, that is, the sum of quantified pairwise
connectivity, which we denote as P(G) p(u, v)
u,vCxVV
for graph G. Apparently P(G) is maximized at (n) when G
is a strongly connected graph.
Based on this, we study the following problems::
Definition 1: (Edge disruptor) Given 0 < 3 < 0, a subset
S C E in G (V, E) is a 3edge disruptor if the overall
pairwise connectivity in the G[E\S], obtained by removing
S from G, is no more than /3(). By minimizing the cost of
such edges in S, we further have the /edge disruptor problem,
i.e., find a minimized 3edge disruptor in a strongly connected
graph G(V, E).
Recall that G is strongly connected iff for every vertex v
in G, there is a directed path from v to all other vertices.
A subgraph of G is called a strongly connected component
(SCC) iff it is a maximal subgraph of G with all vertex pairs
u, v within it connected by directed paths in both directions.
Assume that a 3edge disruptor disrupts the connectivity in
G(V, E) by separating it into several smaller SCCs, say C,
G(V, E)
for i = 1... I i.e. V = C,. We have:
i=1
P(G)
2 1
~n
I )
S i=1
 Var(C)
1 1
where Var(C) (C, ) (Ci )2. Therefore,
i=1 i=1
the two key factors affecting pairwise connectivity are the
number of SCCs and the variance of their sizes. They provide
us an alternative measure for evaluating the balance and
fragmentation of the network.
Similarly, we define 3vertex disruptor and its correspond
ing optimization problem:
/3vertex disruptor problem: Given a strongly connected
graph G(V, E) and a fixed number 0 < 3 < 1, find a subset
S C V with the minimum size such that the total pairwise
connectivity in G[V \ S], obtained by removing S from G, is
no more than /3(). Such a set S is called 3vertex disruptor.
III. HARDNESS RESULTS
In this section we show that both the /edge disruptor
and 3vertex disruptor in directed graph are NPcomplete
that denies the existence of polynomial time exact algorithms
unless P = NP. We state a stronger result that both problems are
NPcomplete even in undirected graph with unit cost edges.
Note that only in this section we consider the problem for
undirected graph G(V, E). All results in other sections are
studied on directed graphs, thus solving both homogeneous
and heterogeneous networks.
A. NPcompleteness of 3edge disruptor
The proof involves a reduction from the balanced cut
problem.
Definition 2: A cut (S, V\S) corresponding to a subset S E
V in G is the set of edges with exactly one endpoint in S.
The cost of a cut is the sum of its edges' costs (or simply
its cardinality in the case all edges have unit costs). We often
denote V\S by S.
Finding a min cut in the graph is polynomial solvable
[14]. However, if one asks for a somewhat "balanced" cut of
minimum size, the problem becomes intractable. A balanced
cut is defined as following:
Definition 3: (Balanced cut) Let f be a function from the
positive integers to the positive reals. An fbalanced cut of a
graph G(V, E) asks us to find a cut (S, S) with the minimum
size such that IS, IS\ > f(lVM).
Abusing notations, for 0 < c < 1, we also use cbalanced
cut to find the cut (S, S) with the minimum size such that
min1{S, IS} > clV We will use in our proofs the following
results on balanced cut shown in [15]:
Corollary 1: (Monotony) Let g be a function with the
property
0 < g(n) g(n 1)< 1
'i 2 t'rX2 C2
Fig. 2. Construction of H(VH, EH) from G(V, E)
Then f(n) < g(n) for all n, implies fbalanced cut is
polynomially reducible to gbalanced cut.
Corollary 2: (Upper bound) an6balanced cut is NP
complete for a, c > 0.
It follows from Corollaries 1 and 2 that for every f O(an6)
fbalanced cut is NPcomplete. We are ready to prove the NP
completeness of /edge disruptor:
Theorem 1: (/3edge disruptor NPcompleteness) 3edge
disruptor in undirected graph is NPcomplete even if all edges
have unit weights.
Proof: We prove the result for the special case when 3
2. For other values of 3 the proof can go through with a
slight modification of the reduction. We consider n to be a
large enough number in our proof, say n > 103.
Consider the decision version of the problem that asks
whether an undirected graph G(V,E) contains a edge
disruptor of a specified size:
1 1
ED = {(G, K) I G has a edge disruptor of size K}
2 2
To show that ED is in NPcomplete we must show that it is
in NP and that all NPproblems are polynomial time reducible
to it. The first part is easy; given a candidate subset of edges,
we can easily check in polynomial time if it is a 3edge
disruptor of size K. To prove the second part, we show that
fbalanced cut is polynomial time reducible to ED where
n ^2L["]+ n
f L 2 +72
Let G(V, E) be a graph in which one seeks to find a f
balanced cut of size k. Now construct the following graph
H(VH, EH): VH V U C1 U C2 where V' = v i e V }
and C1, C2 are two cliques of size ["L]. The total number of
nodes in H is, hence, N = 2 [L + n. Beside edges inside
two cliques we put an edge (vi, vj) for each edge (i,j) e E.
In addition, connect each vertex vi to [L + 1 vertices in Ci
and [L] +1 vertices in C2 so that degree difference of nodes
in the cliques are at most one. We illustrate the construction of
H(VH, EH) in Figure 2. We show that there is a fbalanced
cut of size k in G iff H has an 1edge disruptor of size
K n ([ + 1 + k where 0 < k < [L]. Note that
the cost of any cut (S,V\S) in G is at most ISIIV\Sl <
L(IS 2 J L J
In one hand, an fbalanced cut (S, S) of size k in G induces
a cut (Ci U S, C2 U S) with size exactly n ([2 + 1) + k.
a cut T
If we select the cut as the disruptor, the pairwise connectivity
will be at most ( ).
On the other hand, assume that H has an edge disruptor
of size K = n ( + 1) + k. Because separating n nodes
in a clique from the other nodes in that clique requires cutting
at least n( n) > n (L[J + 1) + k edges. Hence, there
are at most n nodes separated from both the cliques. A direct
consequence is that at least ([L ] n) nodes remain connected
in each clique after removing edges in the disruptor. Denote
the sets of those nodes by C' and C' respectively. C' and C'
cannot be connected otherwise the pairwise connectivity will
exceed 1 ( ). Denote by X1, X2 the set of nodes in V' that
are connected to C' and C' respectively. Since, C' and C'
are disconnected we must have Xi n X2 = .
We will modify the disruptor without increasing its size and
the pairwise connectivity such that no nodes in the the cliques
are split. For each u E C \C' remove from the disruptor all
edges connecting u to C' and add to the disruptor all edges
connecting u to X2. This will move u back to the connected
component that contains C, while reducing the size of the
disruptor at least ([J ] n) n. At the same time select an
arbitrary node v E X1 and add to the disruptor all remaining
v's adjacent edges. This increases the size of the disruptor at
most (L[ + 1) + n while making v isolated. By doing so
we decrease the size of the disruptor by (L[J n) n 
((L[J + 1) + n) > 0. In addition, the pairwise connectivity
will not increase when we connect u to C/ at the same time
with disconnecting v from C'. If no nodes left in Xi we can
select v E X2 as in that case IC' U X2I > I C U Xi that
makes sure the pairwise connectivity will not increase. We
repeat the same process for every node in C2\C' also and
note that I(Ci\C') U (C2\C2)\ < n as proved in previous
paragraph. At the end of the process C = C1 and C C2.
We will prove that X1 U X2 V' i.e. (X1, X2) induces
a cut in V(G). Assume not, the cost to separate C1 U X1
from C2 U X2 will be at least ([L ] + 1)(IV' X1I + V 
2) (L ] + 1)(2n IX1I X21) > (Ln +)(n+l) >
([n 2 ] + + t i
n Lj + i) k that is a contradiction.
Since X1 U X V' we have the disruptor induces a cut
in G. To have the pairwise connectivity at most 2 ( ) both
(Ci U X1) and (C2 U X2) must have size at least N !N
If follows that X1 and X2 must have size at least f(n)
n 2L 3+n
[ ]. The cost of the cut induced by (X1, X2) in
G will be n (LJ + 1) + k n( + 1) k.
B. Hardness of 3vertex disruptor
Theorem 2: 3vertex disruptor in undirected graph is NP
complete.
Proof: We ignore the details and present here the sketch
of the proof. We show that vertex cover is polynomial time
reducible to 3vertex disruptor. Recall that a vertex cover of
a graph is a set of vertices such that each edge of the graph
is incident to at least one vetex of the set. The problem of
finding a minimum vertex cover is an NPhard optimization
problem.
Let G(V, E) be a graph in which one seeks to find a vertex
cover of size k. Note that if we remove nodes in a vertex cover
from the graph, the pairwise connectivity in the graph will be
zero. Hence, by setting 0, G has a vertex cover of size k
iff G has an 3vertex disruptor of size k.
One can also avoid using 3 0 by replacing each vertex in
G by a clique of large enough sizes, say O(n), that ensures no
vertices in cliques will be selected in the /vertex disruptor. U
Given the NPhardness of the 3vertex disruptor, the best
possible result is a polynomial time approximation scheme
(PTAS) that given a parameter c > 0, produces a (1 + c)
approximation solution in polynomial time. Unfortunately,
such a scheme does not exist for /vertex disruptor, unless
P = NP.
Theorem 3: Unless P = NP, /vertex disruptor has no
polynomial time approximation scheme.
Proof: In the case 3 = 0, the problem is equivalent to
finding the minimum vertex cover in the graph. In [16], Dinur
and Safra showed that approximating vertex cover within
constant factor less than 1.36 is NPhard. Hence, a PTAS
scheme for /vertex disruptor does not exist unless P = NP. E
IV. APPROXIMATING 3EDGE DISRUPTOR
USING TREE DECOMPOSITION
In this section, we present an O(log2 n) pseudo
approximation algorithm for the /edge disruptor problem in
the case when all edges have uniform cost i.e. c(u, v) =
1 V(u, v) E E(G). Formally, our algorithm finds in an uniform
directed graph G a P'edge disruptor whose the cost is at most
O(log2 n)OPT3 ED, where 4 < 3 < 3' and OPT3 ED is
the cost of an optimal /edge disruptor.
As shown in Algorithm 1, the proposal algorithm consists
of two main steps. First, we constructs a decomposition tree
of G by recursively partition the graph into two halves with
directed cbalanced cut. Second, we solve the problem on the
obtained tree using a dynamic programming algorithm and
transfer this solution to the original graph. These two main
steps are explained in the next two sections.
A. Balanced TreeDecomposition
A tree decomposition of a graph is a recursive partitioning
of the node set into smaller and smaller pieces until each piece
contains only one single node. We show the tree construction
in Algorithm 1 (line 1 to 11). Our decomposition tree is a
rooted binary tree whose leaves represent nodes in G. (Because
our decomposition tree is a binary tree with n leaves, it will
contains exactly n 1 nonleaf nodes. One can prove this with
induction on number of nodes.).
Definition 4: Given a directed graph G(V, E) and a subset
of vertices S C V. We denote the set of edges outgoing from
S by 6+(S); the set of edges incoming to S by 6 (S). A
cut (S, V\S) in G is defined as + (S). A cbalanced cut is
a cut (S,V\S) s.t. min{SI, IV\SI} > clVI. The directed
to
level 3
..... .. ...... .. level 4
/ *
Fig. 3. A part of a decomposition tree. F {t2, t3, t5, t6} is a G
paritionable. The corresponding partition {V(t2), V(t3), V(ts), V(te)} in
G can be obtained by using cuts at ancestors of nodes in F i.e. to, ti, t4.
cbalanced cut problem is to find a cbalanced cut of the
minimum size.
Note that a cut (S, V\S) separate pairs (u, v) E S x (V\S)
as paths from v to u cannot exist i.e. no SCC can contain
vertex in both S and V\S.
The decomposition procedure is as follows. We start with
the tree T containing only one root node to. We associate the
root node to with the vertex set V of G i.e. V(to) V(G).
For each node ti E T whose V(ti) contains more than one
vertex and V(ti) has not been partitioned, we partition the
subgraph G[V(ti)] induced by V(ti) in G using a cbalanced
cut algorithm. In detail, we use the directed cbalanced cut
algorithm presented in [17] that finds in polynomial time a c'
balanced cut within a factor of O( logn) from the the optimal
cbalanced cut for c' = c and fixed constant a. The constant
c is chosen to be 1 Create two child nodes til, L
of ti in T corresponding to two sets of vertices of G[V(ti)]
separated by the cut. We associate with ti a cut cost cost(ti)
equal to the cost of the cut.
We define the root node to to be on level 1. If the a node
is on level 1, all its children are defined to be on level 1 + 1.
Note that all collections of vertices corresponding to nodes
in a same level forms a partition in V i.e. they are pairwise
disjoint sets of V.
One important parameter of the decomposition tree is the
height i.e. the maximum level of nodes in T. Using balanced
cuts guarantees a small height of the tree that in turn leads
to a small approximation ratio. When separating V(ti) using
the balanced cut, the size of the larger part is at most (1 
c') V(ti)1. Hence, we can prove by induction that if a node ti
is on level k, the size of the corresponding collection V(ti) is
at most IVI x (1 c')k1. It follows that the tree's height is
at most ( log ,, n).
B. Algorithm
In this section, we present the second main step which uses
the dynamic programming to search for the right set of nodes
in T such that the cuts to partition those corresponding sets
of vertices in G have the minimum cost and the obtained
pairwise connectivity is at most 3' (,). The details of this step
are shown in Algorithm 1 (lines 12 to 18).
Denote a set F {t,,, t,..., t. } C VT where VT is
the set of vertices in T so that V(t.U), V(tu),.. V(t,,) is
k
a partition of V(G) i.e. V(G) V,, We say such a subset
h=l
F is Gpartionable. Denote by A(ti) the set of ancestors of
t, in T and A(F) J A(ti). It is clear that a F is G
titF
partionable iff F satisfies:
1) Vti,tj e F : ti i A(tj) and tj g A(ti)
2) Vti e VT, ti is a leaf: A(ti) n F p1
In case F is Gpartitionable, we can separate
V(tGl),V(tU),...,V(Lt) in G by performing the
cuts corresponding to ancestors of node in F during
the tree construction. For example in Figure 3, we show a
decomposition tree with a Gpartitionable set {t2,t3,t5,t6}.
The corresponding partition {V(t2),V(ts),V(t5), V(t6)}
in G can be obtained by cutting V(to),V(tl),V(t4)
successively using balanced cuts in the tree construction.
The cut cost, hence, will be cost(to) + cost(tl) + cost(t4).
In general, the total cost of all the cuts to separate
V(t, ), V(t,),...,V(t) will be:
cost(F)= cost(t,)
t A(F)
The pairwise connectivity in G then will be:
P(F)= P(G[V(tL)])
Of course we want to find F so that P(F) < ,'( ) i.e. the
union of cuts to separate V(t, ), V(t ),..., V(t, ) forms a
/'edge disruptor in G. Because of the suboptimal structure
in T, finding such a Gpartitionable subset F in VT with
minimum cost(F) can be done in O(n3) using dynamic
programming as described netx.
Denote cost(ti,p) the minimum cut cost to make the
pairwise connectivity in G[V(ti)] equal to p using only cuts
corresponding to nodes in the subtree rooted at ti. The
minimum cost for a Gpartionable subset F that induces a
/'edge disruptor of G is then min{cost(to,p)  < /'(n)}
where to is the root node in T.
The value of cost(ti,p) can be calculated using following
recursive formula:
0 if P(G[V(tj)]) p
cost(ti,p) in cost(tl, p) +cost(ti2,p2) cost(ti) otherwise
P1 +P2=P
where til, ti are children of ti.
In the first case, when P(G[V(ti)]) < p we cut no edges in
G[V(ti)] hence, cost(t, p) = 0. Otherwise, we try all possible
combinations of pairwise connectivity pi in V(ti1) and p2 in
V(ti2) that sum up to p. The combination with the smallest
cut cost is then selected.
Using the above recursive formula we can find a minimum
cost Gpartitionable subset F that induces a 3'edge disruptor
in G. We now prove that the cost of the 3'edge disruptor
induced from the minimum cost Gpartitionable F found
in the dynamic programming algorithm is no more than
O(log n)Oi'i ED, where Ol'i ED denote the cost of the
Algorithm 1. 3edge Disruptor
Input: Uniform directed graph G = (V, E) and 0 < 3 < 3' < 1
Output: A 3'edge disruptor of G.
/* Construct the decomposition tree */
1. c 1 .
2. T(VT,ET) ({to}, 6), V(to) V(G), 1(to) 1
3. while 3 unvisited ti with IV(ti)l > 2 do
4. Mark ti visited, create new child nodes til, t2 of ti.
5. 1(til), 1(ti2) (ti) + 1.
6. VTVT U{ti,ti2}
7. ET ET U {(ti, ti), (ti, ti2)}
8. Separate G[V(ti)] into two using directed cbalanced cut.
9. Assign two obtained partitions to V(til), V(ti2)
10. cost(ti) The cost of the balanced cut
11. end while
/* Find the minimum cost Gpartitionable */
12. for ti E T in reversed BFS order from root node to do
13. forp 0 to /3' ()
14. if P(G[V(ti)]) < p then
15. cost(ti,p) 0
16. else
17. cost(ti,p) min{cost(til,Pl)+
cost(ti2,p2) + cost(ti) I P1 +p2 p}
18. Find F with P(F) = min{cost(to,p) I P < /3'(')}
19. Return union of cuts used at A(F) during tree construction
Algorithm 2. Find a good Gpartitionable subset of T
that induces a 3'edge disruptor in G
1: XT c4
2: for t, 6 T in BFS order from to
3: if (3. E Xp : IV(t) n Cil > (1 c)lV(t)l)
4: and (A(t.) n XT = ) then
5: XT XT U{t}
optimal 3edge disruptor in G. It follows directly from the
following lemma:
Lemma 1: There exists a Gpartitionable subset of T that
induces a 3'edge disruptor whose cost is no more than
0 log 2n) Oli ED
Proof: Denote Dp an optimal 3edge disruptor in G.
Removing D3 from G we obtain a set of SCCs, say Ca =
{C1, C2, ...,Ck}.
We construct a Gpartitionable subset XT as in the Algo
rithm 2. We visit nodes in T in a topdown manner i.e. every
parent must be visited before its children. This can be done
by visiting nodes in Bread First Search (BFS) order from the
root node to. For each node ti if there exists some component
Cj e C3 that V(ti) contains more than (1 c)V(ti) nodes in
Cj (all leaves in T satisfies this condition) and no ancestors
of ti have been ever selected into XT, then we select ti as a
member of XT.
One can verify that the obtained XT satisfies two mentioned
conditions of a Gpartitionable subset. We are going to prove
that P(XT) < P'(:)
P (X,) < V(t)
tiCXT
1
2 (t(t
0 cC V(ti)nc 03 (1c)V(ti)
1
2
V(ti)
\v(ti)12
2
EC )nC
Cj cc v(ti)n c, >31 v (ti)
2
S1
,z ;09) 2
1311 (1: 2 _) < 3,(n
Finally we show that cost(XT) < O(log2 n)Oi'i ED. Let
denote by h(T) the height of T and L' the set of nodes at
level i in TG. We have:
h(T)
cost(XT) = cost(tL) (1)
i=1 t C(LinA(XT))
If t, E A(XT) then t, is not selected to XT. Hence, there
exists Cj E C so that V(t,) n C < (1 c) V(t,) (otherwise
t, was selected into XT as it satisfied the conditions in the
line 3, Algorithm 2). To guarantee c < 1 c we constrain
3'
c < 1/2 i.e. /3> .
The edges in the optimal 3edge disruptor Dp separate C,
from the other SCCs. Hence, Dp also separates Cj n V(t,)
from the V(t,)\Cj in G[V(t,)]. Denote sep(t,,Da) the
set of edges in Dp separating Cj n V(t,) from the rest in
G[V(t,)]. Obviously, sep(t,, Dp) is a directed cbalanced cut
of G[V(t,)]. Since, the cut we used in the tree construction
is only 0( logn) times the optimal cbalanced cut. We have
cost(t ) < O( ogn)lsep(t,, DD).
Recall that if two nodes tu,tL are on a same level
then V(tu) and V(t,) are disjoint subsets. It follows that
sep(t,, Dp) and sep(tv, Dp) are also disjoint sets. Therefore,
we have:
tSe(LinA(XT))
cost(t,) < O( logn)
SO( logn)
teU(L nA(XT))
t6(LTnA(X)
Isep(t,, Dp)I
sep(t,, Dp)I
0( logn)Opt3ED
Since h(T) is at most O(log n), it follows from the Equation
1 that cost(XT) < O(log2 n)OptaED. This yields the proof.
Since there exists a Gpartitionable subset of T that
induces a /'edge disruptor whose cost is no more than
O (log2 n) Oli ED as shown in Lemma 1 and the dynamic
programming is always able to find such a set F, the following
Theorem follows immediately.
Theorem 4: Algorithm 1 achieves a pseudoapproximation
ratio of O(log n) for the 3edges disruptor problem.
V. 3VERTEX DISRUPTOR
We present a polynomial time algorithm that finds a 3'
vertex disruptor in the directed graph G(V, E) whose the
size is at most O(log n log log n) times the optimal 3vertex
disruptor where 0 < 3 < 3'2. The algorithm involves in two
phases. In the first phase, we split each vertex v E V into
two vertices v+ and v while putting an edge from v to v+
and show that removing v in G has same effects as removing
edge (v+ v) in the new graph. In the second phase, we
try to decompose the new graph into SCCs capping the sizes
of the largest component while minimizing the number of
removed edges. The constraints on the size of each component
is kept relaxing until the set of cut edges induces a 3'vertex
disruptor in the original graph G. We summarize our algorithm
in Algorithm 3.
Given a directed graph G(V, E) for which we want to find
a small P'vertex disruptor, we split each vertex in G into two
new vertices to obtain a new directed graph G'(V', E') where
V' { v,v+ I v V}
E' = {(v v+)  ve V}
u {(a+ v) I (U v) E}
The new graph G'(V', E') will have twice the number of
vertices in G i.e. IV' 2VI 2n. An example for the
first phase is shown in Figure 4.
We set the costs of all edges in E' {(v v+) I v E V}
to 1 and other edges in E' to +o00 so that only edges in Ev
can be selected in an edge disruptor set. In implementation,
it is safe to set the costs of edges not in E' to O(n) noting
that by paying a cost of 2n we can effectively disconnect all
edges in E' and make P(G') 0.
Consider a directed edge disruptor set D' C E' that con
tains only edge in E1'. We have a onetoone correspondence
between D' to a set D, {(v (v v+) e D'} in
G(V, E) which is a vertex disruptor set in G. Since G and
G' have different maximum pairwise connectivity, ( n for
G and (2n 21)2n for G', the fractions of pairwise connectivity
remaining in G and G' after removing D, and D' are,
however, not simply related to each other.
In the second phase of Algorithm 3, we observe that if we
remove edges and separate the graph into SCCs then there is a
correlation between the pairwise connectivity in the remaining
graph and the maximum size of SCC. The smaller the sizes of
SCCs, the smaller pairwise connectivity in the graph. However,
the smaller the maximum size of each SCC, the more edges
are needed to be cut. Hence, we perform binary search to find a
right upper bound for size of each SCC in G'. In the algorithm,
the lower bound and upper bound of the size of each SCC
are PBV'l and PBV' respectively. At each step we try to find
a minimum capacity edge set in G'(V', E') whose removal
partitions the graph into strongly connected components of
size at most B/V', where /3 = 1j x c. We round the
S2( erun h
value of 3 to the nearest multiple of c so that the number of
steps for the binary search is bounded by log _. The problem
of finding a minimum capacity edge set to decompose a graph
of size n into strongly connected components of size at most
pn is known as pseparator problem. We use here the algorithm
presented in [18] that for a fixed e > 0 finds a pseparator in
directed graph G whose value is at most O (T. log n log log n)
times OPt(p )separator where Opt(p )separator is the cost
of the optimal (p c)separator. Finally, we convert the cut
edges in G' to vertices in G to obtain the 3'vertex disruptor.
Lemma 2: Algorithm 3 always terminates with a 3'vertex
disruptor.
Proof: We show that whenever 3 < 3' then the corre
sponding D, found in Algorithm 3 is a /'vertex disruptor
in G. Consider the edge disruptor D' in G' induced by D,.
We first show the mapping between SCCs in G[V\D,] and
SCCs in G'[E'\D'], the graph obtained by removing D' from
G'. Partition the vertex set V of G into: (1) D,: the set
of removed nodes (2) Vingl,: the set of nodes that are not
in any cylcle i.e. they are SCCs of size one (3)Vconn.cted:
union of remaining SCCs that sizes are at least two, say
Vconnected = LJ= Ci, C > 2. Vertices in Vconnected belong
to at least one cycle in G.
We have following corresponding SCCs in G'[E'\D']:
1) v E D, SCCs {v+} and {v }. Since after removing
(v  v+) v+ does not have incoming edges and v
does not have outgoing edges.
2) v E Vingl, SCCs {v+} and {v }. Since v does
not lie on any cycle in G. Assume v+ belong to some
SCC of size at least 2 i.e. v+ lies on some cycle in
G'. Because the only incoming edge to v+ is from v.
It follows that v is preceding v+ on that cycle. Let
u u be the successive vertices of v+ on that cycle.
We have u and v belong to a same SCC in G which
yields a contradiction. Similarly, v cannot lie on any
cycle in G'.
3) SCC C, C Vconected SCC C = {, V+ V C,}.
This can be shown using a similar argument to that in
the case v Vsingle.
Since D' is a /separator, the sizes of SCCs in G'[E'\D']
are at most / 2n. It follows that the sizes of SCCs in G[V\D,]
are bounded by /n. Denote the set of SCCs in G[V\D,] by
C with the convention that vertices in D, become singleton
SCCs in G[V\D,]. Therefore, we have:
P(G[V\D,])
2)2)
CitC
Cj(z1: c
V)
S( E )C IV1)
(cZec /
2 \2]<<)V I) /3, IVI)
This guarantees that the binary search always finds a 3'vertex
disruptor and completes the proof. U
S*v nL
I~ w+ c
(a) (b)w
Fig. 4. Conversion from the node version in a directed graph (a) into the
edge version in a directed graph (b)
Theorem 5: Algorithm 3 always finds a 3'vertex disruptor
whose the size is at most O(log n log log n) times the optimal
3vertex disruptor for 3'2 > 3 > 0.
Proof: It follows from the Lemma 2 that Algorithm 3
terminates with a 3'vertex disruptor D,. At some step the
capacity of D, equals to the capacity of /separator D' in
G' where 3 is at least 3' c according to Lemma 2 and
the binary search scheme. The cost of the separator is at
most 0 (log n log log n) times the Opt( c)separator using
the algorithm in [18].
Consider an optimal (/3'2 9)vertex disruptor D' of G
and its corresponding edge disruptor D' in G'. Denote the cost
of that optimal vertex disruptor by Opt(,2 9)VD If there
exists in G[V\D,] a SCC C, so that IC, > (3' 2c)n then
P(G[V\D]) > ((3' 2c)n 2)((/3' 2c)n 1) > (/32 _
9e)() when n > 20('1. Hence, every SCCs in G'[V\D']
have size at most (3'2e)(2n) i.e. D' is an (/3'2c)separator
in G'. It follows that Opt(,296)VD > OPt(3' 2,)separator
in G'.
Because 3 e > 3' 2e, we have Opt(p)separator <
Opt(,'2c)separator < Opt( Q29.)VD.
The size of the vertex disruptor ID, = D'I is at most
O (log n log log n) times Opt(p )separator Thus, the size
of found 3'vertex disruptor D, is at most O(log n log log n)
times the optimal (3'2 9c)vertex disruptor. As we can choose
arbitrary small c, setting 3 /32 9c completes the proof. U
We further investigate the relation between 3edge disruptor
and 3vertex disruptor on G and G' as shown next.
A. Approximating edge disruptor is at least as hard as ap
proximating vertex disruptor
We show that an approximation algorithm for general edge
disruptor in directed graph also yields an approximation algo
rithm for directed vertex disruptor with the same approxima
tion ratio.
Lemma 3: A 3edge disruptor set in the directed graph G'
induces the same cost 3vertex disruptor set in G.
Proof: We use D, and D' for vertex disruptor in G and
edge disruptor in G'.
Given P(G'[E'\D']) < p(3(2) we need to prove that:
P(G[V\D,]) < P3() where n = V.
Assume G[V\D,] has 1 SCCs of size at least 2, say C,, i =
1... 1. The corresponding SCCs in G'[E'\D']wil be C,, i =
1... I where \C 21C,.
Since (2) (n ) >0, V0 < k < n. We
(2hae (n1)n(2n )
have:
P(G[V\D,])
(2)
(tici 1 (_ <3
i () 2 (
2) 2 ) 7
Algorithm 3. 3'vertex disruptor
Input: Directed graph G = (V, E) and fixed 0 < 3' < 1.
Output: A 3'vertex disruptor of G
1. G'(V', E')  (6, 6)
2. Vv V:V' V U{v ,v }
3. Vv e V: E' E' U{(v v+)},c(v ,v+) 1
4. V(u v) E : E' E' U{u+ * v }, c(u+, v) oo
5. 13 0,  1
6. Dv V(G)
7. while (3 3 > c) do
8. p L j x
9. Find De C E' to separate G' into strongly connected
components of sizes at most P3V' using algorithm in [18]
10. D, {v E V(G) I (v+ v) e De}
11. if P(G[V\D,]) < /3() then
12. /3 3
13.
14.
15. el
16.
Remove nodes from D, as long as P(G[V\D,]) < /3()
if IDvy > ID,l then Dv = D,
se
3=
18.end while
19. Return Dv
Lemma 4: A 3vertex disruptor set in the directed graph
G' induces the same cost (3 + e)edge disruptor set in G for
fixed c > 0.
Proof: We use the same notations in the proof of Lemma
3. Given P(G[V\D,]) < /( 2,) we need to prove that:
P(G'[E'\D']) < (3 + e)(). We have:
P(G'[E'\D'])
2()
CJl(n ICl) P(G[V\D,])
(n 1)n(2n 1) )
P(G[V\D,]) 1 ) 1
() 2n 1 2n+ 1
< 13+e
when n > 1 J + 1. +
Lemmas 3 and 4 provide a path to approximate the 3vertex
disruptor. The fixed c constant in Lemma 4 is inevitable due
to the natural of the problem.
VI. RELATED AND PRIOR WORKS
The related works to be included in this section consist
of three aspects: traditional vulnerability measures, existing
critical vertex/edge detections, and related graph partition
problems.
The classic vulnerability measures are twofold: global
graph measures and local nodal measures. The former ones
are mainly functions of graph properties, e.g., the number
of vertices/edges, operational OD pairs, operational paths,
minimum shortest paths [1] [2] [3] [4]. However, some of these
attributes cannot be calculated in polynomialtime for dense
networks. Plus these functions do not reveal the set of most
critical vertices and edges, thus are not suitable to assess the
network vulnerability in terms of connectivity. The local nodal
measures are mainly based on the centrality of each vertex in
the graph, which consist of degree centrality, betweenness,
closeness, and eigenvector centrality [5]. However, these mea
sures fail to indicate accurate vulnerabilities and cannot reveal
the global damage done on the network under attacks.
Previous works involving with Critical Vertex/Edge defined
it as the minimum number of vertices/edges whose removal
disconnects the graph, regardless how disconnected it is.
Several heuristics have been proposed with no approximation
ratio guaranteed. In the context of wireless network, there
are some DFSbased centralized algorithms [6] [7], pairwise
routingbased distributed algorithms [8] and localized algo
rithms [9]. Regardless of their high communication overheads,
the definition of critical vertex/edge is only limited to elements
disconnecting graphs, thus cannot apply this concept to the
vulnerability evaluation.
Related to our 3disruptor problem is some works along
the line of graph partitioning problems. Specifically, lin
ear programming relaxation based on multicommodity flow
which was provided by Leighton and Rao [19] obtained
an O(logn) approximation algorithm for SPARSEST CUT
and an O(logn) bicriteria approximation algorithm for c
BALANCED CUT; Arora, Rao, Vazirani [20] improved these
results to O( n) using semidefinite relaxation with trian
gle inequality constraints. In addition, divide and conquer
technique via spreading metrics [21] [18] and hierarchical
decomposition tree techniques [22] [23] [24] also become
major tools for graph approximation algorithms. Meanwhile,
many inapproximability results shown in the sequence of
papers [25] [26] and [27] related the minimum bisection to the
unique games conjecture. A variant of the Pvertex disruptor
in undirected network is presented in [28] in which the authors
seek for a set of k nodes whose removal maximally decreases
the connectivity in the graph. However, no approximation
results have been developed in the literature for all variants
of the problems presented in this paper.
VII. CONCLUSION
We introduced a novel model and measurement for evalu
ating the vulnerability of general networks even in the case of
networks with unidirectional links. We also formulated new
optimization problems corresponding to the model, studied
their hardness, and proposed two pseudoapproximation al
gorithms with provable performance bounds. Being the first
quantitative measure directing at global network connectivity,
our method provides an accurate vulnerability assessment
toward different types of network.
For the future work, we are aiming to bridge the gap in the
proposed algorithm of the 3vertex disruptor problem for any
value of 3 < 3' and further tightening the ratio, which we
conjecture that using the semidefinite relaxation in [20], we
may be able to reduce the ratio to O(/ ).
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