Title: Optima
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Title: Optima
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Publication Date: January 2007
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? T I MA
Mathematical Programming Society Newsletter



ISMP 2006 and Rio de Janeiro

.6 *e .. - e.




ISMP 2006 and Rio
de Janeiro
Rio de Janiero welcomed the participants
to the 19th International Symposium on
Mathematical Programming with its world
famous features: beautiful landscape,
friendliness and hospitality of the people,
excellent food and great caipirinhas. The
weather was sunny and beautiful before and
after, but was less inviting during the exact
days of the conference, which we believed
was skillfully arranged by the organizers to
ensure exceptionally high attendance of the
sessions of the Symposium. And indeed it
further improved an exceptional scientific
program, which included outstanding
plenary and semi-plenary talks, great
contributed and invited sessions. Thanks
to the infectious Brazilian style and at the
same time, the exciting (due to being in
Rio against all odds, namely Varig's ill-
timed bankruptcy, after perilous journeys
around the world) atmosphere made
communication and interaction among all
of us extremely pleasant. It is difficult to
capture this atmosphere on paper in a short
time and we resisted trying. However, we
are reproducing for the OPTIMA readers
one of the peaks of the conference which
was, as is traditional for the Mathematical
Programming Symposia, the awarding
of the triennial prizes. We have asked
the chairmen of the committees to
provide detailed descriptions of each prize
which are given below. These interesting
outcome is presented below showing
once again the achievements and exciting
challenges of Mathematical Programming
in the past and in the next few years.
As the new editorial board we hope to
earn a warm welcome from the readers of
OPTIMA by starting our first issue with the
reminder of the good times in Rio.

Alberto Caprara, Andrea Lodi,
Katya Scheinberg

The A. W. Tucker Prize
Description and Committee
The A. W. Tucker Prize is awarded for
an outstanding paper or Ph.D. thesis by
a student in the area of mathematical
programming. This year on the Tucker
Prize committee were Monique Laurent,
Jong-Shi Pang, Ruediger Schultz
and Tom McCormick (chair).



-j-' '
8' SB "

Finalists and Winner
The 2006 prize attracted 21 nominations
(a record number) of amazingly high
quality, most of which were Ph.D.
theses. There was a lot of geographical
diversity in the nominations: 11 from
North America, 7 from Europe, and 3
from elsewhere; also diversity in area of
study: doing some double-counting, 11
were on continuous optimization, 12 on
discrete, 9 on stochastic problems, and
14 included some computation. The
committee was especially pleased to see so

much computation, and that such strong
work is being done in the stochastic area.
The three finalists were Jos6 Rafael
Correa, Dion Gijswijt and Uday V. Shanbhag
and overall, Dr. Shanbhag's impressive
dissertation, by its breath and depth,
qualified the work as the winner of the
2006 A.W. Tucker Prize.

Jos6 Rafael Correa graduated as a
Mathematical Engineer from Universidad
de Chile in 1999. He completed his Ph.D.
in Operations Research at MIT under the




supervision of Michel Goemans and Andreas
Schulz in June 2004. Currently Dr. Correa
is an Assistant Professor at the School of
Business at Universidad Adolfo Ibaiiez.
Dr. Correa was named as a Tucker
Prize finalist for his Ph.D. thesis titled
"Approximation Algorithms for Packing and
Covering Problems". This thesis develops
approximation algorithms for applied
problems in three quite different areas.
The first comes from scheduling packets
in an interconnection network, which
gets abstracted into a problem of coloring
edges in bipartite graphs, and then further
into a bin-packing problem. The second
considers the natural problem of packing
rectangles (or higher-dimensional cubes)
into boxes. It extends previous results about
2-dimensional bin-packing to find an
asymptotic polynomial time approximation
scheme for packing d-dimensional cubes
into unit cubes, and it gets new results for
packing rectangles into square bins. In both
cases new tools are developed to make the
arguments work. The third considers the
classic problem of scheduling precedence-
constrained jobs on a single machine to
minimize the average weighted completion
time. It significantly extends known results
by using linear programming relaxations
to show that essentially all known 2-
approximation algorithms comply with the
"Sidney decomposition", and then shows
that the sequencing problem can be seen as a
special case of vertex cover.

Dion Gijswijt completed his curriculum
in Mathematics at the University of
Amsterdam, where he graduated in
2001, and received his Ph.D. degree
under the supervision of Lex Schrijver
in September 2005. He is currently a
researcher at the Eitvis University in
Budapest, and he will join the University
of Amsterdam in September 2006.
Dr. Gijswijt has been selected as a
Tucker Prize finalist for his Ph.D. thesis
entitled "Matrix Algebras and Semidefinite
Programming Techniques for Codes". This
thesis presents a novel method for bounding
the maximum cardinality of a nonbinary
code with given minimum Hamming
distance, which is one of the most basic
problems in coding theory and an instance
of the stable set problem in Hamming
graphs. The method also applies to the dual
problem of bounding the minimum size of
covering codes. The new bound he proposes
improves the classical linear programming

bound of Delsarte, and gives sharper
estimates than the state-of-the art methods
on many instances of codes up to length 12
on alphabets of sizes 3, 4, and 5.
The method of Dr. Gijswijt relies on deep
insight from noncommutative algebra allied
with the use of semidefinite optimization.
While the algebraic ingredient in the
Delsarte method is the commutative Bose-
Mesner algebra of the Hamming scheme,
the noncommutative Terwilliger algebra
plays a crucial role in Gijswijt's method.
Extending earlier work of Schrijver for the
binary case, Gijswijt finds the explicit block-
diagonalization of the Terwilliger algebra,
which enables him to apply symmetry
reduction and reformulate his new bound
via a compact semidefinite program of size
O(n^3) for a code with word length n.
Gijswijt also studies the link to the matrix-
cut method of Lovisz and Schrijver.
This work shows how sophisticated
algebraic techniques can be successfully used
for exploiting symmetries and formulate
compact semidefinite relaxations for hard
combinatorial optimization problems, thus
adding a new algebraic technique to the
mathematical programming toolbox.

Uday V. Shanbhag obtained his
undergraduate degree in engineering at the
Indian Institute of Technology, Bombay
(Mumbai) in 1993, and his Ph.D. in the
department of Management Science and
Engineering at Stanford University in 2006
under the direction of Walter Murray.
Currently, he is an Assistant Professor
in the Department of Mechanical and
Industrial Engineering at the University
of Illinois at Urbana-Champaign.
Uday Shanbhag's doctoral dissertation,
titled "Decomposition and Sampling
Methods for Stochastic Equilibrium
Problems", deals with a novel class
of extremely difficult yet practically
very important optimization problems
constrained by equilibrium conditions and
subject to uncertainty. Successive chapters
deal with stochastic quadratic programs
with recourse (extending previous works
on importance-sampling-based L-shape
decomposition methods), mathematical
programs with complementarity constraints
(MPCCs) (proposing an interior-point
based method that calculates stationary
solutions satisfying an MPCC second-order
condition, compared to prior methods that
found only first-order solutions), two-stage
MPCCs with uncertainty (solving via a

primal-dual method that relies on sampling
and a scenario-based decomposition), and
a two-period spot-forward market under
uncertainty formulated as a stochastic Nash-
Stackelberg game (motivated by applications
to electric power markets with oligopolies).
The research utilizes and advances the
state-of-the-art nonlinear programming and
Monte Carlo sampling methods for tackling
such problems. Several new ideas and
formulations are introduced; great care is
placed on the highly technical convergence
analysis; and the results from the
implementation of the proposed methods
on realistic applications are interpreted with
interesting insights. The end product is an
outstanding piece of work that combines
theory, algorithms, applications, and
implementations, bringing together and
significantly advancing several areas in
continuous optimization, and enabling the
application of new optimization paradigms.

The Beale-Orchard-Hays Prize
Description and Committee
The Beale-Orchard-Hays committee
composed by Bill Cook, Michael Juenger,
Franz Rendl, and Steve Wright (chair)
received 8 nominations in various
areas of computational mathematical
programming. According to its character,
the prize is awarded to a paper appearing
within the past three years that
describes development of software, the
computational evaluation and testing of
new algorithms, and the development
of new methods for empirical testing of
algorithms, all in the area of mathematical
programming. The nominations spanned
a broad spectrum of the discipline and
included both well established and new
software packages, along with testing
of new algorithmic approaches.

The prize was awarded to Nick Sahinidis
and Mohit Tawarmalani for their paper
"A polyhedral branch-and-cut approach
to global optimization", Mathematical
Programming, Series B 103 (2005), pp. 225-
249. The approaches described in this paper
are implemented in the BARON system
which represents a powerful approach
for the global optimization of nonlinear
optimization problems, including problems
with integer variables. The nominated paper
develops techniques that enhance previous
versions of BARON. In particular, it uses

10S T I MA73



factorable decompositions of nonlinear
functions into subexpressions to construct
polyhedral outer approximations that
exploit convexity more thoroughly and
yield tighter underestimators. BARON also
incorporates techniques from automatic
differentiation, interval arithmetic, and
other areas to yield an automatic, modular,
and relatively efficient solver for the very
difficult area of global optimization.

The George B. Dantzig Prize
Description and Committee
The George B. Dantzig prize is awarded
jointly by the Mathematical Programming
Society and the Society for Industrial and
Applied Mathematics. The prize is awarded
for original research, which by its originality,
breadth and depth, is having a major impact
on the field of mathematical programming.
The contributions) for which the award
is made must be publicly available and
may belong to any aspect of mathematical
programming in its broadest sense. Strong
preference is given to candidates that have
not reached their 50th birthday in the year
of the award. Committee Members this year
were Arkadi Nemirovskii, Lex Schrijver,
Jong-Shi Pang and Bob Bixby (chair).

The committee has selected Eva Tardos
as the lone recipient of the Dantzig Prize
this year. She is warded the prize for
deep and wide-ranging contributions to
mathematical programming. In the 1980s,
she solved a long-standing open problem by
finding the first strongly polynomial-time
algorithm for minimum-cost flows, a truly
ground-breaking result. Most subsequent
work on minimum-cost flows, including
several of the currently fastest algorithms
has roots in her revolutionary method.
This first result in network flow is just one
of numerous results by Tardos in the area.
More generally, a wide range of network
flow models can be viewed as a special
case of linear programming, and hence is
solvable in polynomial time. Tardos has
made significant advances in the design of
more efficient polynomial-time algorithms
for these problems by exploiting their
network structure. Among these is the first
polynomial-time combinatorial algorithm
for generalized network flows. She also
obtained polynomial-time algorithms for
certain multi-commodity flow problems;

among them the maximum concurrent
flow problem. These results have led
to further generalizations for a wide
array of combinatorially-defined linear
programs, generally known as fractional
packing and covering problems.
Tardos has been a leader in the use of
sophisticated mathematical programming
techniques in the design of approximation
algorithms for NP-hard discrete
optimization problems. For example,
for the generalized assignment problem,
Tardos showed that any feasible solution
to a natural linear programming relaxation
can be rounded to be integer, where the
resulting solution cost not greater than
twice that of the fractional solution.
Throughout the years, Eva Tardos has
renewed herself scientifically by changing
the focus of her work, most recently by
laying foundations for new, important
directions in algorithmic game theory.
Her work showing performance bounds
for "selfish routing" has gained attention
for its crystalization of the notion of the
price of anarchy. One statement of her
breakthrough result is that the negative
effect of allowing users to selfishly route
their traffic is completely offset by building
a network of double the capacity.
Tardos is a stimulating lecturer and
collaborator, always willing to exchange
ideas, and often with clever and
surprisingly ingenious approaches that
turn out to be basis for subsequent
research. She has trained an impressive
line of students, written many valuable
surveys, and has played a particularly
important role in linking Mathematical
Programming and Computer Science.
The depth, breadth, originality, and
impact of her work make her a very
deserving winner of the Dantzig Prize.

The Fulkerson Prize
Description and Committee
The Fulkerson Prize is for outstanding
papers in the area of discrete mathematics.
The prize is sponsored jointly by the
Mathematical Programming Society and
the American Mathematical Society. Up
to three awards are presented at each
(triennial) International Symposium of the
Mathematical Programming Society. To
be eligible for this year's award, a paper
had to be published in a journal between
January 2000 and December 2005. The

committee consisted of Noga Alon, Bill
Cunningham and Michel Goemans (chair).

Three prizes were given this year and
the three winning papers are:

Manindra Agrawal, Neeraj Kayal and
Nitin Saxena, "PRIMES is in P", Annals
of Mathematics, 160 (2004), pp. 781--793.

Testing whether an integer is a prime
number is one of the most fundamental
computational and mathematical problems.
The existence of short certificates for both
compositeness and primality was known
since the 70's and suggested that primality
testing might be in P. Yet, despite numerous
efforts and a flurry of algorithms, it was not
until 2002 that Agrawal, Kayal and Saxena
devised the first deterministic polynomial-
time algorithm for primality testing.
Earlier algorithms had either assumed the
generalized Riemann hypothesis, or were
randomized or were only subexponential.
This is a stunning development. This
result is a true masterpiece, combining
algebraic and number theoretic
results in a seemingly simple way.

Mark Jerrum, Alistair Sinclair and Eric
Vigoda, "A polynomial-time approximation
algorithm for the permanent of a
matrix with nonnegative entries", J.
ofACM, 51 (2004), pp. 671--697.

The permanent of a matrix has been
studied for over two centuries, and is
of particular importance to statistical
physicists as it is central to the dimer and
Ising models. For a 0-1 matrix, it represents
the number of perfect matching in the
corresponding bipartite graph. Although
polynomial-time computable for planar
graphs, the computation of the permanent
is #P-complete for general graphs as shown
by Valiant in 1979. This opened the
search for approximation schemes. In this
paper, Jerrum, Sinclair and Vigoda give
the first Fully Polynomial Randomized
Approximation Scheme for computing
the permanent of any 0-1 matrix or any
nonnegative matrix. This is a remarkable
result. Their algorithm is based on
updating a Markov chain in a way that
quickly converges to a rapidly mixing
non-uniform Markov chain on perfect

0SP T I MA73



matching and near-perfect matching.
Their work builds upon the earlier
pioneering work of Jerrum and Sinclair
who initiated the use of rapidly mixing
Markov chains for combinatorial problems.

Neil Robertson and Paul D. Seymour,
"Graph Minors. XX. Wagner's Conjecture",
Journal of Combinatorial Theory,
Series B, 92(2004), pp. 325--357.

Kuratowski's theorem says that a graph is
planar if and only if it does not contain K(
or K3,3 as a minor. Several other excluded
minor characterizations are known, and
Wagner conjectured that any minor-closed
graph property can be characterized by a
finite list of excluded minors. Restated, this
says that for any infinite family of finite
graphs, one of its members is a minor of
another one. In a remarkable tour de force,
Robertson and Seymour proved Wagner's
conjecture, and this paper appeared as part
20 of their monumental work on the theory
of graph minors. Their proof of the Graph
Minor Theorem required the development
of many graph theoretic concepts, such
as linkages and tree-width. This is a
spectacular achievement in graph theory
with far reaching consequences. It shows,
for example, that embeddability in any fixed
surface can be characterized by a finite list
of excluded minors, or that the disjoint
paths problem can be solved in polynomial
time for a fixed number of terminals.

The Lagrange Prize
Description and Committee
The Lagrange Prize in Continuous
Optimization, given jointly by
Mathematical Programming Society
and Society for Industrial and Applied
Mathematics, is awarded for outstanding
works in the area of continuous
optimization. The committee for the 2006
prize consisted of John Dennis, Nick Gould,
Adrian Lewis, and Mike Todd (chair).

A number of strong nominations were
received, and the committee had a lively
discussion concerning their merits. The
committee unanimously chose Roger
Fletcher, Sven Leyffer and Philippe
Toint to win the prize for their papers:

Roger Fletcher and Sven Leyffer,
"Nonlinear programming without
a penalty function", Mathematical
Programming, 91 (2002), pp.239-269

Roger Fletcher, Sven Leyffer, and Philippe
L. Toint, "On the global convergence
of a filter-SQP algorithm", SIAM J.
Optimization, 13 (2002), pp. 44-59.

The citation reads: In the development of
nonlinear programming over the last decade,
an outstanding new idea has been the
introduction of the filter. This new approach

to balancing feasibility and optimality has
been quickly picked up by other researchers,
spurring the analysis and development of
a number of optimization algorithms in
such diverse contexts as constrained and
unconstrained nonlinear optimization,
solving systems of nonlinear equations, and
derivative-free optimization. The generality
of the filter idea allows its use, for example,
in trust region and line search methods,
as well as in active set and interior point
frameworks. Currently, some of the most
effective nonlinear optimization codes are
based on filter methods. The importance of
the work cited here will continue to grow as
more algorithms and codes are developed.
The filter sequential quadratic
programming (SQP) method is proposed
in the first of the two cited papers. Many
of the key ideas that form the bases of later
non-SQP implementations and analyses
are motivated and developed. The paper
includes extensive numerical results, which
attest to the potential of the algorithm.
The second paper complements the
first, using novel techniques to provide a
satisfying proof of correctness for the filter
approach in its original SQP context. The
earlier algorithm is simplified, and in so
doing the analysis plays its natural role with
respect to algorithmic design.

International conference

on Algorithmic Operations


The 2nd International conference on Algorithmic
Operations Research (AlgOR 2007) will be
held at Simon Fraser University-Surrey, Canada
during January 21-23, 2007. To submit a paper for
presentation or to register for the conference please
use the online registration form at the conference web
page. An international Operations Research Case
Competition for students is also planned. For further
details please visit the conference web page at:


Selected papers from conference will be published as
a special issue of the journal Algorithmic Operations
Research (http://journals.hil.unb.ca/index.php/AOR)

CIME School on Nonlinear Optimization
preliminary announcement

Organizes a school on
Nonlinear Optimization
The school will take place in
Cetraro,Cosenza, Italy Grand Hotel
S. Michele from July 1 to July 7, 2007.

Prof. Immanuel Bomze
Univ. of Vienna, Austria
Global Optimization

Prof. Vladimir Demianov
St. Petersbourg State Univ., Russia
Non-smooth optimization

Prof. Roger Fletcher
Univ. of Dundee, UK
Sequential Quadratic Programming

Prof. Tamas Terlaky
Mc Master Univ. Canada
Interior Point Methods

Course directors are
Prof. Gianni Di Pillo
Univ. La Sapienza, Rome, Italy

Prof. Fabio Schoen
Univ. di Firenze, Italy

Registration begins January
2007. It is free, but is required
in advance. Visit CIME at www.
cime.unifi.it for registration.

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Match, match, match,

and match again

Gerhard J. \' .: ,ii i "

We discuss four discrete problems, one
problem for each occurrence of the word
match' in the title of this paper. The
solutions of all four problems are based
on underlying matching problems.

Matching is one of the most fundamental,
most popular, and most studied problems
in Math-ematical Programming. Every
OPTIMA reader knows that the goal of a
matching problem is speaking somewhat
sloppily to pair up objects so that the
sum of the weights of these pairs becomes
maximum (or minimum). Every OPTIMA
reader knows everything that can possibly be
known about perfect matching, maximum
cardinality matching, bipartite matching,
aug-menting paths, the Hungarian
method, and the blossom algorithm of
Edmonds. (And if thereshould exist some
reader who does not know everything,
he might want consult the survey [7]by
Gerards or the books [9] by Schrijver.)
In this paper we will discuss four
discrete problems that can be solved quite
elegantlythrough the classical matching
machinery. At first sight, however, none of
these four problems appears to be a 'clean'
matching problem:

* Problem #1 is a scheduling problem.
The matched objects are jobs, and
feasible match ings must agree with
an underlying partial order.
Problem #2 is a geometric problem.
The matched objects are points, and
in feasible matching certain pairs of
edges must not show up together. (In
general, matching problems with such
forbidden pairs are NP-complete.)
Problem #3 again is a geometric
problem. The matched objects are
points and wedges, and feasible
matching must satisfy a global
covering condition. (In general,
matching problems with an additional

global condition are NP-complete.)
Problem #4 concerns a two-player
game. The game is centered around
paths, and not around matching.

All four problems can be rewritten and
reformulated, so that they turn into
standard matching problems. All four
problems are easy to grasp, and they
all possess nice and short solutions.
Hence, these problems might be
appropriate for spicing up homework
assignments and class- room exercises.

1 Scheduling under precedence
Fujii, Kasami & Ninomiya [3] consider a
processing system with two machines and
n jobs. Each of the jobs must be processed
for one time unit without interruption
on one of the two machines. The jobs
are partially ordered by a transitive, anti-
symmetric, irreflexive precedence relation:
Ifi - the processing of job i must be completed
before the processing of job j can start.
Two jobs i andj are called compatible, if
neither i - andj can be processed simultaneously on
the two machines without violating the
precedence constraints. A job without
predecessors is called a minimaljob. The
problem is to decide whether all jobs can
be processed within t time units. Without
loss of generality, we will assume that
there are exactly n = 2t jobs. (If n > 2t, the
problem trivially has no solution. If n < 2t,
then we may add 2t n dummy jobs to the
instance that do not interact with the other
jobs through the precedence constraints.)
Here is a solution for this problem. We
construct an undirected auxiliary graph
G with the jobs as vertices, and an edge
between job i and jobj if and only if i and
j are compatible. If there exists a feasible
schedule, then the pairs of jobs that are
processed simultaneously yield a perfect
matching in the auxiliary graph G. The
surprise is that also the reverse statement is
true: If G contains a perfect matching, then
there exists a feasible schedule.

Lemma 1 If the auxiliary graph G has some
perfect matching, then it also has a perfect
matching that matches two minimal jobs.

*gwoegi@win.tue.nl. Department of Mathematics and Computer Science, TU Eindhoven, PO.

Proof. Put a cost of 2 on every edge
that connects two minimal jobs, and
put a cost of 1 on all the other edges in
G. Compute a perfect matching /M of
maximum cost. Suppose for the sake of
contradiction that /M does not contain any
edges that connect two minimal jobs.
Consider a minimal job a and its partner
b in M/; since b is non-minimal, it is
preceded by some other minimal job a'.
Repeated application of this observation
yields that there exists a cyclic sequence
al, ... a., a, a = al of minimal jobs with
non-minimal partners b,, .... b, b = b,
in M4, such that a -< b, holds for all i = 1, .
s. Now consider the four jobs a, bi, ,,
and bi+ for some fixed i.

If b. and b.i are compatible, then
we could replace the edges [a, b,]
and [ai+1, bi,] in 4M by [a, a+l]
and [b,, b,+,]. This would increase
the cost of M4; a contradiction.
If b -< bi, holds, then a+ -< b.
together with the transitivity of the
precedence constraints implies ai+1
ai+-< bi+1. Then ai+ and bi, are not
compatible; a contradiction.

Therefore, bi+1 A bi must hold for i =
1; : : : ; n. But this means that the jobs
bl; : : : ; bs; bl form a closed cycle in the
precedence relation! This contradiction
completes the argument. m

Now it is obvious how to get a polynomial
time algorithm for the scheduling
problem: We apply Lemma 1 to -nd
a perfect matching that matches two
minimal jobs i andj. We schedule i
andj into the earliest empty time slot,
remove them from G, and we repeat this
procedure until G becomes empty.
Of course there are faster (and more
complex) algorithms for solving this two-
machine scheduling problem. The fastest
known algorithm has a running time linear
in the number of jobs and the number of
precedences; it is due to Gabow & Tarjan
[4]. The corresponding problem for three
machines is an outstanding open question
in complexity theory; see Garey & Johnson
[6]. All natural generalizations of the above
approach to three machines turn out to
be incorrect or do not work in polynomial

Box 513, 5600 MB Eindhoven, The Netherlands.

0SP T I MA73



2 Crossing-free matching
The following problem was one of the
twelve problems posed at the 1979 'i
Lowell Putnam Mathematical Competition
(that's the earliest reference that I could find,
although I suspect the problem to be much
much older than that): Consider a set R of n
red points and a set B of n blue points in the
Euclidean plane, such that no three points
of R u B lie on a common line. Show:
There exist n pairwise disjoint, straight line
segments that match the blue points to the
red points.
And here is the solution: Consider a
matching 4M between the point sets R and B
that minimizes the total Euclidean length of
the corresponding n straight line segments.
Suppose that 4M contains two line segments
R1B1 and R2B2 that intersect in a common
point X. The triangle inequality yields |RX|
+ IXB2I > IRB2I and IR2X| + IXB, > IR2B|,
which leads to

> IRB2| + IRaBJ.

Thus, by switching the partners of R1 and
R2 in M4 one could decrease the total length
of S4. Therefore, 4M is crossing-free.
Hershberger & Suri [8] construct
crossing-free matching in O(n log n) time.
Their approach is purely based on geometric
observations, and does not use graph theory.
Their time complexity O(n log n) is best
possible, since the problem contains sorting
as a special case.

3 Illuminating the ocean
In the following problem, the reader should
think of the plane as an ocean, of the points
P, as lighthouses in this ocean, and of the
wedges Wk as regions that are illuminated
by floodlights: There are n points P, ....
, P in the Euclidean plane. Furthermore,
there are n rays that emanate from the
origin and cut the Euclidean plane into n
wedges W, . .,W These wedges span
angles a,, ... a( with 'k 1 ak = 27.
Show: The wedges W, . ., W can be
translated from the origin to the points P ,
..., Pn (one wedge per point), such that
their translates again cover the entire plane.
The special case of this problem with n = 4
points and with angles ak i /2 was posed
at the 1967 All Soviet Union Mathematical

Competition. We will describe a solution of
stunning beauty for the general problem.
It is due to Galperin & Galperin [5],
and it is based on an auxiliary matching
problem. Let k, denote the vector of length
1/ o .,(Tr '2) in the direction of the angle-
bisector of W. We write (ia,b)to denote the
inner product of the two vectors and b,
and we write I||I|| to denote the length of
vector a b. Translating wedge Wk from the
origin to some other anchor point P yields
the region W[P].

Lemma 2 Consider two points P and
Q, and two integers k, I with 1 < k; I
< n. If Q lies inside Wk[P] but outside
W[P],then (PPQ, v ) > ( .

Proof. Let l denote the angle between
PQ and k. Then Q e W%[P] yields
IP\ < ak/2, which implies cos(p)
> cos(a,/2). Whence

A symmetric argument centered around
the angle between P Q and vector v
yields that (P Q, vi) < \IP Q. .

We now fix some arbitrary point Q in the
plane, and we define the Q-cost of assigning
wedge Wk to point P. as (P Q, vk ). We
compute a matching 4M between wedges
and points that maximizes the total Q-cost.
By renumbering the points, we may assume
that for k = 1, ... ,n matching 4M assigns
wedge Wk to point Pk, and that hence the
translated wedges are Wk[P]. Here is the
first beautiful observation:

Lemma 3 If matching M maximizes
the Q-cost, then point Q is covered
by one of the translated wedges.

Proof. Suppose not. Observe that for every
point P., there exists some wedge Wk with Q
E W,[P]; we denote this situation by
i -> k. Clearly, the relation contains some
directed cycle c1 -> C2 * c c+1
= c for some s > 2. But then the following
cyclic switch would increase the Q-cost of
M4: For k = 1, . s re-assign wedge Wk+l
from point Pck+ to point Pck By Lemma 2,
this contradicts the maximality of M4. m

Here is the second beautiful observation:
The Q-costs do depend on the choice of

Q, but the optimal matching 4M does not.
Indeed, let it be some assignment of wedges
to points and consider two points Q and
Q2. Then the difference between the two
objective values of iT under Q1-costs and
under Q2-costs equals nX
Since this difference is independent of the



(Q0k= 1, f ))


Pk- Z 2,v(k)
PkQ2, V(k) =

(QjQ, 6E)

assignment 7T, matching 4M maximizes the
Q-cost for every possible point Q. Hence,
by Lemma 3 every possible point Qis
covered by one of the translated wedges.

4 A path-forming game
Player 1 and Player 2 play the following
game on an undirected graph G: They
alternately select the next vertex of a
simple path P in G. Player 1 is free to
select the starting vertex of P in his
first move. The first player unable to
move loses the game. Who wins this
game, if both players play optimally?
We will show that Player 1 has a winning
strategy if and only if G has no perfect
matching. It is easy to see that Player 2 has a
winning strategy, if G does contain a perfect
matching M4: Whenever Player 1 moves to
some vertex v, then Player 2 reacts by simply
moving to the partner vertex of v in 4M.
And Player 1 wins the game, if G does not
contain a perfect matching: Player 1 fixes
an arbitrary maximum cardinality matching
4M (that by assumption is non-perfect), and
then starts the path in a vertex u'that is not
covered by M4. Whenever Player 2 moves
to some vertex v, Player 1 reacts by moving
to the partner vertex of v in M4. Note that
Player 2 must always move to a vertex
u" that is covered by M4; otherwise, the
augmenting path P from u'to u"could be
used to increase the cardinality of S4.
This game shows up (for instance)
as an exercise in Bondy & Murty [2].
Interestingly, Bodlaender [1] has proved that
the corresponding path-forming game in
directed graphs is PSPACE-complete, and
hence intractable.

10S T I MA73

B R<(PQ ) =cos(() |hPQ|| | ||
IR1B + |R2B2B PQ Pk) = cos( k I2) I > IP QI :
=(IR1X + IXB ) + (IR2X + IXB I) cos)
( / YI Y I\/ I( YI IYR I\




[1] H.L. Bodlaender (1993).
Complexity of path-forming
games. Theoretical Computer
Science 110, 215{245.

[2] J.A. Bondy and U.S.R. Murty
(1976). Graph Theory with
Applications. Elsevier/
MacMillan, New York.

[3] M. Fujii, T. Kasami, and K.
Ninomiya (1969). Optimal
sequencing of two equivalent
processors. SIAM Journal
of Applied Mathematics 17,

[4] H.N. Gabow and R.E.
Tarjan (1985). A linear-time
algorithm for a special case
of disjoint set union. Journal
of Computer and System
Sciences 30, 209{221.

[5] V. Galperin and G. Galperin
(1981). Osveschenije
ploskosti prozhektorami
(Illuminating a plane with
spotlights). Kvant 11, 28{30.
(In Russian).

[6] M.R. Garey and D.S. Johnson
(1979). Computers and
Intractability: A Guide to the
Theory of NP-Completeness.
Freeman, San Francisco.

[7] A.M.H. Gerards (1995).
Matching. In: Network
Models, M.O. Ball, T.L.
Magnanti, C.L. Monma,
and G.L. Nemhauser (eds.),
Handbooks in Operations
Research and Management
Science, Volume 7, 135{224.

[8] J. Hershberger and S. Suri
(1992). Applications of a
semi-dynamic convex hull
algorithm. BIT 32, 249{267.

[9] A. Schrijver (2003).
Combinatorial Optimization:
Polyhedra and Eciency.
Springer, Berlin, Heidelberg.

Symposium in honor of George Nemhauser

A two-day symposium will be held in honor
of George's 70th birthday. George has had
a profound impact on many people in
the operations research and mathematical
programming community. He was
president of MPS from 1989-1992. This
event will provide the participants with an
opportunity to show their appreciation. The
symposium will include a small number
of scientific presentations on general
topics and also special presentations that

cover George's work and his contributions
through various stages of his career.
Speakers at the symposium include:

Cindy Barnhart
Bob Bixby
Bill Cook
Gerard Cornuejols
Marshall Fisher
Rob Garfinkel
Martin Groetschel
John Jarvis

Tom Magnanti
Bill Pulleyblank
Don Ratliff
David Ryan
Mike Todd
Mike Trick
Laurence Wolsey

The Symposium will be held at Georgia
Tech, start at noon on Thursday,
July 26, 2007 and end at 1:30 on
Friday, July 27 (George's birthday is
the 27th). There will be a banquet on
Thursday evening. The registration
fee is $100. Further announcements
and registration details will follow.
Mike Ball e Martin Savelsbergh

Application for Membership

I wish to enroll as a member of the Society.
My subscription is for my personal use and not for the benefit of any library or institution.
O I will pay my membership dues on receipt of your invoice.
E I wish to pay by credit card (Master/Euro or Visa).





Mail to:
Mathematical Programming Society
3600 University City Sciences Center
Philadelphia, PA 19104-2688 USA

Cheques or money orders should be
made payable to The Mathematical
Programming Society, Inc. Dues for
2007, including subscription to the
journal Mathematical Programming,
are US 80. Retired are 50.
Student applications: Dues are 25.
Have a faculty member verify your
student status and send application
with dues to above address.

Faculty verifying status


0 PT MA 3






Peter Ladislaw Hammer
December 23,1936 December 27, 2006

Peter Ladislaw Hammer was born in
Timisoara, Romania, on December 23,
1936. He earned his Ph.D. in mathematics
under Academician Grigore C. Moisil at
the University of Bucharest in 1966. He
defected to Israel in 1967 where he became
a professor at the Technion in Haifa. After
moving to Canada, he taught from 1969
to 1972 at the University of Montreal,
and from 1972 to 1983 at the University
of Waterloo. In 1983, he moved to the
USA and became a professor at Rutgers
University, where he founded RUTCOR
- the Rutgers Center for Operations
Research. He remained the director of
RUTCOR until his untimely death in a
tragic car accident, on December 27, 2006.
For more than 40 years, Peter Hammer
has ranked among the most influential
researchers in the fields of operations
research and discrete mathematics. He
has made numerous major contributions
to these fields, launching several new
research directions. His results have
influenced hundreds of colleagues and
have made a lasting impact on many areas
of mathematics, computer science, and
Most of Peter Hammer's scientific
production has its roots in the work of
George Boole on propositional logic. More
than anyone else, Peter Hammer has used
and extended Boole's machine universalis
to handle questions relating to decision
making, analysis and synthesis as they arise
in natural, economic and social sciences.
Over the span of his scientific career, he
has conducted eclectic forays into the
interactions between Boolean methods,
optimization, and combinatorial analysis,
while adapting his investigations to the most
recent advances of mathematical knowledge
and of various fields of application.
Among the main research topics which
have received his attention, one finds an
impressive array of methodological studies
dealing with combinatorial optimization,
some excursions into logistics and game
theory, numerous contributions to graph
theory, to the algorithmic aspects of
propositional logic, to artificial intelligence
and, more recently, to the development of
innovative data mining techniques. His

publications include 19 books and over 240
scientific papers. (See the Web site rutcor.
rutgers.edu for a complete bibliography.)
At the very onset of his career, as a
researcher at the Institute of Mathematics
of the Academia of Romania, Peter
Hammer wrote several important articles
on transportation problems, jointly with
Egon Balas. At the same time his advisor,
Grigore Moisil, directed him to the study
of Boolean algebra. In this field, a central
role is played by functions depending on
binary variables, and taking either binary
values (i.e., Boolean functions) or real values
(i.e., pseudo-Boolean functions). In a series
of papers, Peter Hammer demonstrated
that a large variety of relevant problems
of operations research, combinatorics and
computer science can be reduced to the
optimization of a pseudo-Boolean function
under constraints described by a system
of pseudo-Boolean inequalities. A further
main conceptual step in his work was
the characterization of the set of feasible
solutions of the above system as solutions of
a single Boolean equation (or, equivalently,
of a satisfiability problem). This led him, in
joint work with Ivo Rosenberg and Sergiu
Rudeanu, to the development of an original
approach inspired from classical Boolean
methods for the solution of a large variety of
discrete optimization problems.
This research project culminated in
1968 with the publication of the book
Boolean Methods in Operations Research
and Related Areas (Springer-Verlag, 1968),
co-authored by Sergiu Rudeanu. This
landmark monograph, which founded the
field of pseudo-Boolean optimization, has
influenced several generations of students
and researchers, and is now considered a
"classic" in Operations Research.
In a sense, Peter Hammer's early
work can be viewed as a forerunner of
subsequent developments in the theory
of computational complexity, since it
was in effect demonstrating that a large
class of combinatorial optimization
problems is reducible to the solution of
Boolean equations. However, this purely
reductionistt" view of his work would
be quite narrow. In fact, Peter Hammer
has systematically used the "canonical"
representation of various problems in terms

of Boolean functions or Boolean equations
to investigate the underlying structure,
the "essence" of the problems themselves.
More than often, this goal is met through
a simplifying process based, once again, on
the tools of Boolean algebra. This approach
provides, for instance, a simple way to
demonstrate that every system of linear
inequalities in binary variables is equivalent
to a set of inequalities involving only 0,1,-1
coefficients, as observed in a joint paper by
Frieda Granot and Peter Hammer (1972). It
also led Peter Hammer, Ellis Johnson and
Uri Peled (1975) to early investigations into
the facial structure of knapsack polyhedra.
In a related stream of research, Peter
Hammer has established numerous fruitful
links between graph theory and Boolean
functions. In a famous joint paper with
VaSek Chvital on the aggregation of
inequalities in integer programming (1977),
he introduced and characterized the class
of threshold graphs, inspired by threshold
Boolean functions. Threshold graphs have
subsequently been the subject of scores of
articles and of a book by Mahadev and
Uri Peled, two of Peter Hammer's former
doctoral students. Other links between
graphs and Boolean or pseudo-Boolean
functions have been explored in joint work
with Claude Benzaken, Dominique de
Werra, Stephan Foldes, Toshihide Ibaraki,
Alex Kelmans, Vadim Lozin, Freddric
Maffray, Bruno Simeone, etc.
Quadratic 0-1 optimization has been
one of Peter Hammer's main fields of
investigation. The theory of roof-duality
(1984), jointly developed with Pierre
Hansen and Bruno Simeone, builds on
concepts from linear programming (linear
relaxations), Boolean theory (quadratic
Boolean equations) and networks
(maximum network flow problems) to
compute best linear approximations of
quadratic pseudo-Boolean functions and
tight bounds on the maximum value of such
functions. Further research along similar
lines has been conducted by Peter Hammer
in collaboration with Endre Boros, Jean-
Marie Bourjolly, Yves Crama, David Rader,
Gabriel Tavares, X. Sun, etc.
Peter Hammer has also shown interest
for the application of Boolean models in

10S T I MA73



artificial intelligence and related fields, as
witnessed by numerous joint papers with
Gabriela and Sorin Alexe, Martin Anthony,
Tiberius Bonates, Endre Boros, Yves Crama,
Oya Ekin, Toshi Ibaraki, Alex Kogan,
Miguel Lejeune, Irina Lozina, and other
collaborators. His contributions bear on
automatic theorem proving, compression
of knowledge bases, algorithms for special
classes of satisfiability problems, etc. About
20 years ago, he launched an innovative
approach to data mining based on a blend
of Boolean techniques and combinatorial
optimization. The basic tenets of this
approach were presented in a joint paper
with Yves Crama and Toshihide Ibaraki
(1988) and were subsequently developed
by Peter Hammer and his coworkers into
a new broad area of research, which he
dubbed Logical Analysis of Data, or LAD
for short. The effectiveness of the LAD
methodology has been validated by many
successful applications to real-life data
analysis problems. In particular, some front-
of-the line medical centers are increasingly
using LAD in the actual practice of medical
diagnosis for a variety of syndromes.
Many aspects of Peter Hammer's
immense contribution to the study of
Boolean functions and their combinatorial
structure are to be found in a forthcoming
monograph entitled Boolean Functions:
Theory, Algorithms, and Applications, co-
authored by Yves Crama and several other
close collaborators of Peter Hammer, to be
published by Cambridge University Press in
Beside his scientific production, Peter
Hammer will undoubtedly be remembered
for his vigorous contribution to and
promotion of discrete mathematics and
operations research. He was the founder

and editor-in-chief of several highly-
rated professional journals, including
Discrete Mathematics, Discrete Applied
Mathematics, Discrete Optimization,
Annals of Discrete Mathematics, Annals
of Operations Research and the SIAM
Monographs on Discrete Mathematics and
Applications. At Rutgers University, Peter
Hammer was the founding Director of the
operations research programme, and he was
largely responsible for developing RUTCOR
into an internationally recognized center
of excellence and an open institute, where
seminars, workshops, graduate courses, and
a constant flow of visitors create a buzzing
and stimulating research environment. He
was also a tireless organizer of professional
conferences and workshops, where he always
made sure to provide opportunities for
interactions between experienced scientists
and younger researchers.
The importance of Peter Hammer's
scientific contribution was acknowledged
by the award of numerous international
distinctions, including the "George
Tzitzeica" prize of the Romanian Academy
of Science (1966), the Euler Medal of
the Institute of Combinatorics and
its Applications (1999), and honorary
degrees from the Swiss Federal Institute
of Technology in Lausanne (1986), the
University of Rome "La Sapienza" (1998),
and the University of Liege (1999). He was
a Fellow of the American Association for
the Advancement of Science since 1974,
and a Founding Fellow of the Institute
of Combinatorics and its Applications.
Several conferences were organized in his
honor, including the First International
Colloquium on Pseudo-Boolean
Optimization (Chexbres, Switzerland,
1987), the Workshop and Symposia

Honoring Peter L. Hammer (Caesarea
Rothchild Institute, University of Haifa,
2003), and the International Conference
on Graphs and Optimization (GO V,
Leukerbad, Switzerland, 2006).
Peter Hammer was not only an
outstanding scholar and a tireless organizer,
but also a kind, generous and humorous
human being. He relished the interaction
with students and colleagues, and made
everybody feel comfortable to work with
him, be it on a mathematical question
(which he was always keen to formulate) or
on planning a conference. He supervised
numerous graduate students with respect
and fatherly understanding, considering
each one of them as his "best student". He
was also a true "citizen of the world": born
in Romania from a Hungarian family, he
subsequently took the Canadian citizenship,
then the US one, wrote joint papers with
co-authors of 28 different countries, fluently
spoke 6 languages (or more), travelled the
world extensively, spent extended periods of
time in Belgium, France, Israel, Italy, Russia,
Switzerland and many other countries, and
developed an extended network of friends
and coworkers on all continents.
Finally, last but certainly not least, Peter
Hammer was a loving husband, father and
grandfather. He is survived by his wife,
Anca Ivanescu, whom he married in 1961
and whose family name he assumed for a
few years, by his two sons Alexander and
Maxim, and by four beloved grandchildren,
Isabelle, Madeline, Annelise, and Oliver.
He will be missed by everyone who knew
him, always and forever.

Endre Boros1, Yves Crama2
and Bruno Simeone3

0SP T I MA73

1 RUTCOR, Rutgers Center for Operations Research, Piscataway, NJ 08854-8003, USA
2 HEC Management School, University of Liege, 4000 Liege, Belgium
3 Department of Statistics, Probability and Applied Statistics, University La Sapienza, 00185 Rome, Italy



MPS-SIAM Book Series on

Michael L. Overton, Editor-in-Chief
Courant Institute of Mathematical Sciences

The goal of the series is to publish a broad range of
titles in the field of optimization and mathematical
programming, characterized by the highest scientific


Variational Analysis in Sobolev and BV Spaces:
Applications to PDEs and Optimization
Hedy Attouch, Giuseppe Buttazzo, and G6rard Michaille
2005 xii + 634 pages Softcover
ISBN- 13:978-0-898716-00-9 ISBN-10:0-89871-600-4
List Price $140.00 MPS/SIAM Member Price $98.00 Order Code MP06

Applications of Stochastic Programming
Edited by Stein WWallace and William T. Ziemba
2005 xvi + 709 pages Softcover
ISBN- 13:978-0-898715-55-2 ISBN- 10: 0-89871-555-5
List Price $142.00 MPS/SIAM Member Price $99.40 Order Code MP05

The Sharpest Cut:
The Impact of Manfred Padberg and His Work
Edited by Martin Grotschel
2004 xi + 380 pages Hardcover
ISBN- 13:978-0-898715-52-1 ISBN- 10: 0-89871-552-0
List Price 99.00 MPS/SIAM Member Price $69.30 Order Code MP04

A Mathematical View of Interior-Point Methods
in Convex Optimization
James Renegar
2001 viii + 117 pages Softcover
ISBN-13: 978-0-898715-02-6 ISBN-10:0-89871-502-4
List Price $43.00 MPS/SIAM Member Price $30.10 Order Code MP03

Lectures on Modern Convex Optimization:
Analysis, Algorithms, and Engineering Applications
Aharon Ben-Tal and Arkadi Nemirovski
2001 xvi + 488 pages Softcover
ISBN- 13:978-0-898714-91-3 ISBN- 10:0-89871-491 -5
List Price $113.50 MPS/SIAM Member Price $79.45 Order Code MP02

Trust-Region Methods
A. R. Conn, N. I. M. Gould, and Ph. L.Toint
2000 xx + 959 pages Hardcover
ISBN- 13:978-0-898714-60-9 ISBN- 10:0-89871-460-5
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SIAM publishes quality
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p..n 11




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DEIS University of Bologna,
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