73 o N O PTIMA Mathematical Programming Society Newsletter January 2007

ISMP 2006 and Rio de Janeiro

73 Prizes 2 Match, match, match, and match again 6 Peter Hammer Obituary 9 op t i m a 7 3 January 2007 page 

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. Finalists and Winner much computation, and that such strong Alberto Caprara, Andrea Lodi, The 2006 prize attracted 21 nominations work is being done in the stochastic area. Katya Scheinberg (a record number) of amazingly high The three finalists were José Rafael Correa Dion Gijswijt Uday V. Shanbhag The A. W. Tucker Prize quality, most of which were Ph.D. , and theses. There was a lot of geographical and overall, Dr. Shanbhag’s impressive Description and Committee diversity in the nominations: 11 from dissertation, by its breath and depth, The A. W. Tucker Prize is awarded for North America, 7 from Europe, and 3 qualified the work as the winner of the an outstanding paper or Ph.D. thesis by from elsewhere; also diversity in area of 2006 A.W. Tucker Prize. a student in the area of mathematical study: doing some double-counting, 11 programming. This year on the Tucker were on continuous optimization, 12 on José Rafael Correa graduated as a Prize committee were Monique Laurent, discrete, 9 on stochastic problems, and Mathematical Engineer from Universidad Jong-Shi Pang, Ruediger Schultz 14 included some computation. The de Chile in 1999. He completed his Ph.D. and Tom McCormick (chair). committee was especially pleased to see so in Operations Research at MIT under the op t i m a 7 3 January 2007 page 

supervision of Michel Goemans and Andreas bound of Delsarte, and gives sharper primal-dual method that relies on sampling Schulz in June 2004. Currently Dr. Correa estimates than the state-of-the art methods and a scenario-based decomposition), and is an Assistant Professor at the School of on many instances of codes up to length 12 a two-period spot-forward market under Business at Universidad Adolfo Ibáñez. on alphabets of sizes 3, 4, and 5. uncertainty formulated as a stochastic Nash- Dr. Correa was named as a Tucker The method of Dr. Gijswijt relies on deep Stackelberg game (motivated by applications Prize finalist for his Ph.D. thesis titled insight from noncommutative allied to electric power markets with oligopolies). "Approximation Algorithms for Packing and with the use of semidefinite optimization. The research utilizes and advances the Covering Problems". This thesis develops While the algebraic ingredient in the state-of-the-art nonlinear programming and approximation algorithms for applied Delsarte method is the commutative Bose- Monte Carlo sampling methods for tackling problems in three quite different areas. Mesner algebra of the Hamming scheme, such problems. Several new ideas and The first comes from scheduling packets the noncommutative Terwilliger algebra formulations are introduced; great care is in an interconnection network, which plays a crucial role in Gijswijt's method. placed on the highly technical convergence gets abstracted into a problem of coloring Extending earlier work of Schrijver for the analysis; and the results from the edges in bipartite graphs, and then further binary case, Gijswijt finds the explicit block- implementation of the proposed methods into a bin-packing problem. The second diagonalization of the Terwilliger algebra, on realistic applications are interpreted with considers the natural problem of packing which enables him to apply symmetry interesting insights. The end product is an rectangles (or higher-dimensional cubes) reduction and reformulate his new bound outstanding piece of work that combines into boxes. It extends previous results about via a compact semidefinite program of size theory, algorithms, applications, and 2-dimensional bin-packing to find an O(n^3) for a code with word length n. implementations, bringing together and asymptotic polynomial time approximation Gijswijt also studies the link to the matrix- significantly advancing several areas in scheme for packing d-dimensional cubes cut method of Lovász and Schrijver. continuous optimization, and enabling the into unit cubes, and it gets new results for This work shows how sophisticated application of new optimization paradigms. packing rectangles into square bins. In both algebraic techniques can be successfully used cases new tools are developed to make the for exploiting symmetries and formulate The Beale-Orchard-Hays Prize arguments work. The third considers the compact semidefinite relaxations for hard Description and Committee classic problem of scheduling precedence- combinatorial optimization problems, thus The Beale-Orchard-Hays committee adding a new algebraic technique to the constrained jobs on a single machine to composed by Bill Cook, Michael Juenger, minimize the average weighted completion mathematical programming toolbox. Franz Rendl, and Steve Wright (chair) time. It significantly extends known results received 8 nominations in various by using linear programming relaxations Uday V. Shanbhag obtained his areas of computational mathematical to show that essentially all known 2- undergraduate degree in engineering at the programming. According to its character, approximation algorithms comply with the Indian Institute of Technology, Bombay the prize is awarded to a paper appearing "Sidney decomposition", and then shows (Mumbai) in 1993, and his Ph.D. in the within the past three years that that the sequencing problem can be seen as a department of Management Science and describes development of software, the special case of vertex cover. Engineering at Stanford University in 2006 computational evaluation and testing of under the direction of Walter Murray. new algorithms, and the development Dion Gijswijt completed his curriculum Currently, he is an Assistant Professor of new methods for empirical testing of in at the University of in the Department of Mechanical and algorithms, all in the area of mathematical Amsterdam, where he graduated in Industrial Engineering at the University programming. The nominations spanned 2001, and received his Ph.D. degree of Illinois at Urbana-Champaign. a broad spectrum of the discipline and under the supervision of Lex Schrijver Uday Shanbhag's doctoral dissertation, included both well established and new in September 2005. He is currently a titled "Decomposition and Sampling software packages, along with testing researcher at the Eötvös University in Methods for Stochastic Equilibrium of new algorithmic approaches. Budapest, and he will join the University Problems", deals with a novel class of Amsterdam in September 2006. of extremely difficult yet practically Winner Dr. Gijswijt has been selected as a very important optimization problems Nick Sahinidis Tucker Prize finalist for his Ph.D. thesis constrained by equilibrium conditions and The prize was awarded to Mohit Tawarmalani entitled "Matrix and Semidefinite subject to uncertainty. Successive chapters and for their paper Programming Techniques for Codes". This deal with stochastic quadratic programs "A polyhedral branch-and-cut approach thesis presents a novel method for bounding with recourse (extending previous works to global optimization", Mathematical the maximum cardinality of a nonbinary on importance-sampling-based L-shape Programming, Series B 103 (2005), pp. 225- code with given minimum Hamming decomposition methods), mathematical 249. The approaches described in this paper distance, which is one of the most basic programs with complementarity constraints are implemented in the BARON system problems in coding theory and an instance (MPCCs) (proposing an interior-point which represents a powerful approach of the stable set problem in Hamming based method that calculates stationary for the global optimization of nonlinear graphs. The method also applies to the dual solutions satisfying an MPCC second-order optimization problems, including problems problem of bounding the minimum size of condition, compared to prior methods that with integer variables. The nominated paper covering codes. The new bound he proposes found only first-order solutions), two-stage develops techniques that enhance previous improves the classical linear programming MPCCs with uncertainty (solving via a versions of BARON. In particular, it uses op t i m a 7 3 January 2007 page 

factorable decompositions of nonlinear among them the maximum concurrent committee consisted of Noga Alon, Bill functions into subexpressions to construct flow problem. These results have led Cunningham and Michel Goemans (chair). polyhedral outer approximations that to further generalizations for a wide exploit convexity more thoroughly and array of combinatorially-defined linear Winners yield tighter underestimators. BARON also programs, generally known as fractional Three prizes were given this year and incorporates techniques from automatic packing and covering problems. the three winning papers are: differentiation, interval arithmetic, and Tardos has been a leader in the use of other areas to yield an automatic, modular, sophisticated mathematical programming Manindra Agrawal, Neeraj Kayal and and relatively efficient solver for the very techniques in the design of approximation Nitin Saxena, "PRIMES is in P", Annals difficult area of global optimization. algorithms for NP-hard discrete of Mathematics, 160 (2004), pp. 781--793. optimization problems. For example, The George B. Dantzig Prize for the generalized assignment problem, Testing whether an integer is a prime Description and Committee Tardos showed that any feasible solution number is one of the most fundamental The George B. Dantzig prize is awarded to a natural linear programming relaxation computational and mathematical problems. jointly by the Mathematical Programming can be rounded to be integer, where the The existence of short certificates for both Society and the Society for Industrial and resulting solution cost not greater than compositeness and primality was known Applied Mathematics. The prize is awarded twice that of the fractional solution. since the 70's and suggested that primality for original research, which by its originality, Throughout the years, Eva Tardos has testing might be in P. Yet, despite numerous breadth and depth, is having a major impact renewed herself scientifically by changing efforts and a flurry of algorithms, it was not on the field of mathematical programming. the focus of her work, most recently by until 2002 that Agrawal, Kayal and Saxena The contribution(s) for which the award laying foundations for new, important devised the first deterministic polynomial- is made must be publicly available and directions in algorithmic game theory. time algorithm for primality testing. may belong to any aspect of mathematical Her work showing performance bounds Earlier algorithms had either assumed the programming in its broadest sense. Strong for "selfish routing" has gained attention generalized Riemann hypothesis, or were preference is given to candidates that have for its crystalization of the notion of the randomized or were only subexponential. not reached their 50th birthday in the year price of anarchy. One statement of her This is a stunning development. This of the award. Committee Members this year breakthrough result is that the negative result is a true masterpiece, combining were Arkadi Nemirovskii, Lex Schrijver, effect of allowing users to selfishly route algebraic and number theoretic Jong-Shi Pang and Bob Bixby (chair). their traffic is completely offset by building results in a seemingly simple way. a network of double the capacity. Winner Tardos is a stimulating lecturer and Mark Jerrum, Alistair Sinclair and Eric The committee has selected Eva Tardos collaborator, always willing to exchange Vigoda, “A polynomial-time approximation as the lone recipient of the Dantzig Prize ideas, and often with clever and algorithm for the permanent of a this year. She is warded the prize for surprisingly ingenious approaches that matrix with nonnegative entries”, J. deep and wide-ranging contributions to turn out to be basis for subsequent of ACM, 51 (2004), pp. 671--697. mathematical programming. In the 1980s, research. She has trained an impressive she solved a long-standing open problem by line of students, written many valuable The permanent of a matrix has been finding the first strongly polynomial-time surveys, and has played a particularly studied for over two centuries, and is algorithm for minimum-cost flows, a truly important role in linking Mathematical of particular importance to statistical ground-breaking result. Most subsequent Programming and Science. physicists as it is central to the dimer and work on minimum-cost flows, including The depth, breadth, originality, and Ising models. For a 0-1 matrix, it represents several of the currently fastest algorithms impact of her work make her a very the number of perfect matchings in the has roots in her revolutionary method. deserving winner of the Dantzig Prize. corresponding bipartite graph. Although This first result in network flow is just one polynomial-time computable for planar of numerous results by Tardos in the area. The Fulkerson Prize graphs, the computation of the permanent More generally, a wide range of network Description and Committee is #P-complete for general graphs as shown flow models can be viewed as a special The Fulkerson Prize is for outstanding by Valiant in 1979. This opened the case of linear programming, and hence is papers in the area of discrete mathematics. search for approximation schemes. In this solvable in polynomial time. Tardos has The prize is sponsored jointly by the paper, Jerrum, Sinclair and Vigoda give made significant advances in the design of Mathematical Programming Society and the first Fully Polynomial Randomized more efficient polynomial-time algorithms the American Mathematical Society. Up Approximation Scheme for computing for these problems by exploiting their to three awards are presented at each the permanent of any 0-1 matrix or any network structure. Among these is the first (triennial) International Symposium of the nonnegative matrix. This is a remarkable polynomial-time combinatorial algorithm Mathematical Programming Society. To result. Their algorithm is based on for generalized network flows. She also be eligible for this year's award, a paper updating a Markov chain in a way that obtained polynomial-time algorithms for had to be published in a journal between quickly converges to a rapidly mixing certain multi-commodity flow problems; January 2000 and December 2005. The non-uniform Markov chain on perfect op t i m a 7 3 January 2007 page 

matchings and near-perfect matchings. The Lagrange Prize to balancing feasibility and optimality has Their work builds upon the earlier Description and Committee been quickly picked up by other researchers, pioneering work of Jerrum and Sinclair spurring the analysis and development of who initiated the use of rapidly mixing The Lagrange Prize in Continuous a number of optimization algorithms in Markov chains for combinatorial problems. Optimization, given jointly by such diverse contexts as constrained and Mathematical Programming Society unconstrained nonlinear optimization, Neil Robertson and Paul D. Seymour, and Society for Industrial and Applied solving systems of nonlinear equations, and “Graph Minors. XX. Wagner's Conjecture”, Mathematics, is awarded for outstanding derivative-free optimization. The generality Journal of Combinatorial Theory, works in the area of continuous of the filter idea allows its use, for example, Series B, 92(2004), pp. 325--357. optimization. The committee for the 2006 in trust region and line search methods, prize consisted of John Dennis, Nick Gould, as well as in active set and interior point Kuratowski's theorem says that a graph is Adrian Lewis, and Mike Todd (chair). frameworks. Currently, some of the most

planar if and only if it does not contain K5 Winners effective nonlinear optimization codes are or K as a minor. Several other excluded based on filter methods. The importance of {3,3} A number of strong nominations were minor characterizations are known, and received, and the committee had a lively the work cited here will continue to grow as Wagner conjectured that any minor-closed discussion concerning their merits. The more algorithms and codes are developed. The filter sequential quadratic graph property can be characterized by a committee unanimously chose Roger programming (SQP) method is proposed finite list of excluded minors. Restated, this Fletcher, Sven Leyffer and Philippe in the first of the two cited papers. Many says that for any infinite family of finite Toint to win the prize for their papers: graphs, one of its members is a minor of of the key ideas that form the bases of later non-SQP implementations and analyses another one. In a remarkable tour de force, Roger Fletcher and Sven Leyffer, are motivated and developed. The paper Robertson and Seymour proved Wagner's “Nonlinear programming without includes extensive numerical results, which conjecture, and this paper appeared as part a penalty function”, Mathematical attest to the potential of the algorithm. 20 of their monumental work on the theory Programming, 91 (2002), pp.239-269 of graph minors. Their proof of the Graph The second paper complements the first, using novel techniques to provide a Minor Theorem required the development Roger Fletcher, Sven Leyffer, and Philippe satisfying proof of correctness for the filter of many graph theoretic concepts, such L. Toint, “On the global convergence as linkages and tree-width. This is a approach in its original SQP context. The of a filter-SQP algorithm”, SIAM J. earlier algorithm is simplified, and in so spectacular achievement in graph theory Optimization, 13 (2002), pp. 44-59. with far reaching consequences. It shows, doing the analysis plays its natural role with respect to algorithmic design. for example, that embeddability in any fixed The citation reads: In the development of surface can be characterized by a finite list nonlinear programming over the last decade, of excluded minors, or that the disjoint an outstanding new idea has been the paths problem can be solved in polynomial introduction of the filter. This new approach time for a fixed number of terminals.

CIME School on Nonlinear Optimization International conference preliminary announcement on Algorithmic Operations Organizes a school on Prof. Tamas Terlaky Research Nonlinear Optimization Mc Master Univ. The school will take place in Interior Point Methods The 2nd International conference on Algorithmic Cetraro,Cosenza, Italy - Grand Hotel Operations Research (AlgOR 2007) will be S. Michele - from July 1 to July 7, 2007. Course directors are held at Simon Fraser University-Surrey, Canada Prof. Gianni Di Pillo during January 21-23, 2007. To submit a paper for Lectures: Univ. La Sapienza, , Italy presentation or to register for the conference please Prof. Immanuel Bomze [email protected] use the online registration form at the conference web Univ. of Vienna, Austria page. An international Operations Research Case Global Optimization Prof. Fabio Schoen Competition for students is also planned. For further Univ. di Firenze, Italy details please visit the conference web page at: Prof. Vladimir Demianov [email protected] St. Petersbourg State Univ., Russia http://math-optima1.surrey.sfu.ca/algor2007/orc.htm Non-smooth optimization Registration begins January 2007. It is free, but is required Selected papers from conference will be published as Prof. Roger Fletcher in advance. Visit CIME at www. a special issue of the journal Algorithmic Operations Univ. of Dundee, UK cime.unifi.it for registration. Research (http://journals.hil.unb.ca/index.php/AOR) Sequential Quadratic Programming op t i m a 7 3 January 2007 page 

global condition are NP-complete.) Proof. Put a cost of 2 on every edge Match, match, match, • Problem #4 concerns a two-player that connects two minimal jobs, and and match again game. The game is centered around put a cost of 1 on all the other edges in paths, and not around matchings. G. Compute a perfect matching M of maximum cost. Suppose for the sake of Gerhard J. Woeginger* All four problems can be rewritten and contradiction that M does not contain any reformulated, so that they turn into edges that connect two minimal jobs. Abstract standard matching problems. All four Consider a minimal job a and its partner We discuss four discrete problems, one problems are easy to grasp, and they b in M; since b is non-minimal, it is problem for each occurrence of the word all possess nice and short solutions. preceded by some other minimal job a'. 'match' in the title of this paper. The Hence, these problems might be Repeated application of this observation solutions of all four problems are based appropriate for spicing up homework yields that there exists a cyclic sequence a a a a on underlying matching problems. assignments and class- room exercises. 1, . . . , s, s+1 = 1 of minimal jobs with non-minimal partners b1, . . . , bs, bs+1 = b1 M a b i Introduction 1 Scheduling under precedence in , such that i+1 p i holds for all = 1, . constraints . . , s. Now consider the four jobs ai, bi, ai+1, Matching is one of the most fundamental, and bi+1 for some fixed i. most popular, and most studied problems Fujii, Kasami & Ninomiya [3] consider a processing system with two machines and in Math-ematical Programming. Every • If bi and bi are compatible, then n jobs. Each of the jobs must be processed +1 OPTIMA reader knows that the goal of a we could replace the edges [ai, bi] matching problem is – speaking somewhat for one time unit without interruption and [ai+1, bi+1] in M by [ai, ai+1] sloppily – to pair up objects so that the on one of the two machines. The jobs and [bi, bi+1]. This would increase sum of the weights of these pairs becomes are partially ordered by a transitive, anti- the cost of M; a contradiction. maximum (or minimum). Every OPTIMA symmetric, irreflexive precedence relation: • If bi p bi+1 holds, then ai+1 p bi reader knows everything that can possibly be If i p j holds for two jobs i and j, then together with the transitivity of the known about perfect matching, maximum the processing of job i must be completed precedence constraints implies ai+1 cardinality matching, bipartite matching, before the processing of job j can start. ai+1p bi+1. Then ai+1 and bi+1 are not aug-menting paths, the Hungarian Two jobs i and j are called compatible, if compatible; a contradiction. method, and the blossom algorithm of neither i p j nor j p i holds, that is, if i Edmonds. (And if thereshould exist some and j can be processed simultaneously on Therefore, bi+1 Á bi must hold for i = reader who does not know everything, the two machines without violating the 1; : : : ; n. But this means that the jobs he might want consult the survey [7]by precedence constraints. A job without b1; : : : ; bs; b1 form a closed cycle in the Gerards or the books [9] by Schrijver.) predecessors is called a minimal job. The precedence relation! This contradiction problem is to decide whether all jobs can In this paper we will discuss four completes the argument.  discrete problems that can be solved quite be processed within t time units. Without elegantlythrough the classical matching loss of generality, we will assume that Now it is obvious how to get a polynomial machinery. At first sight, however, none of there are exactly n = 2t jobs. (If n > 2t, the time algorithm for the scheduling these four problems appears to be a 'clean' problem trivially has no solution. If n < 2t, problem: We apply Lemma 1 to ¯nd matching problem: then we may add 2t - n dummy jobs to the a perfect matching that matches two instance that do not interact with the other minimal jobs i and j. We schedule i • Problem #1 is a scheduling problem. jobs through the precedence constraints.) and j into the earliest empty time slot, The matched objects are jobs, and Here is a solution for this problem. We remove them from G, and we repeat this feasible match ings must agree with construct an undirected auxiliary graph procedure until G becomes empty. an underlying partial order. G with the jobs as vertices, and an edge Of course there are faster (and more • Problem #2 is a geometric problem. between job i and job j if and only if i and complex) algorithms for solving this two- The matched objects are points, and j are compatible. If there exists a feasible machine scheduling problem. The fastest in feasible matchings certain pairs of schedule, then the pairs of jobs that are known algorithm has a running time linear edges must not show up together. (In processed simultaneously yield a perfect in the number of jobs and the number of general, matching problems with such matching in the auxiliary graph G. The precedences; it is due to Gabow & Tarjan forbidden pairs are NP-complete.) surprise is that also the reverse statement is [4]. The corresponding problem for three • Problem #3 again is a geometric true: If G contains a perfect matching, then machines is an outstanding open question problem. The matched objects are there exists a feasible schedule. in complexity theory; see Garey & Johnson points and wedges, and feasible [6]. All natural generalizations of the above matchings must satisfy a global Lemma 1 If the auxiliary graph G has some approach to three machines turn out to covering condition. (In general, perfect matching, then it also has a perfect be incorrect or do not work in polynomial matching problems with an additional matching that matches two minimal jobs. time.

*[email protected]. Department of Mathematics and , TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. op t i m a 7 3 January 2007 page 

2 Crossing-free matchings Competition. We will describe a solution of Q, but the optimal matching M does not. stunning beauty for the general problem. Indeed, let π be some assignment of wedges The following problem was one of the It is due to Galperin & Galperin [5], to points and consider two points Q and twelve problems posed at the 1979 William 1 and it is based on an auxiliary matching Q2. Then the difference between the two Lowell Putnam Mathematical Competition → problem. Let vk denote the vector of length objective values of π under Q1-costs and (that's the earliest reference that I could find, 1/ cos(π /2) in the direction of the angle- k → under Q2-costs equals nX although I suspect the problem to be much bisector of W . We write 〈a→,b 〉to denote the k → Since this difference is independent of the R n → much older than that): Consider a set of inner product of the two vectors a and b, B n → red points and a set of blue points in the and we write ||a || to denote the length of → Euclidean plane, such that no three points vector a→ b. Translating wedge W from the R ∪ B k of lie on a common line. Show: origin to some other anchor point P yields n There exist pairwise disjoint, straight line the region W [P]. segments that match the blue points to the k red points. Lemma 2 Consider two points P and And here is the solution: Consider a Q, and two integers k, l with 1 ≤ k; l matching M between the point sets R and B ≤ n. If Q lies inside W [P] but outside → k → that minimizes the total Euclidean length of → → W [P], then 〈P Q, v 〉 > 〈P Q, v 〉. the corresponding n straight line segments. l k l assignment π, matching M maximizes the Suppose that M contains two line segments Q-cost for every possible point Q. Hence, Proof. Let β denote the angle between → Q R B and R B that intersect in a common → by Lemma 3 every possible point is 1 1 2 2 P Q and v . Then Q ∈ W [P] yields point X. The triangle inequality yields |R X| k k covered by one of the translated wedges. 1 |β| ≤ α /2, which implies cos(β) + |XB | > |R B | and |R X| + |XB | > |R B |, k 2 1 2 2 1 2 1 ≥ cos(α /2). Whence which leads to k 4 A path-forming game → → 〈P Q, v→ 〉 = cos(β) · ||P Q|| · ||v→ || Player 1 and Player 2 play the following |R B | + |R B | k → →k 1 1 2 2 = cos(β) · ||P Q|| / cos(α /2) ≥ ||P Q||: game on an undirected graph G: They = (|R X| + |XB |) + (|R X| + |XB |) k 1 1 2 2 alternately select the next vertex of a = (|R X| + |XB |) + (|R X| + |XB |) 1 2 2 1 A symmetric argument centered around simple path P in G. Player 1 is free to > |R B | + |R B |. → 1 2 1 1 the angle between P Q and vector v→ select the starting vertex of P in his → → l yields that 〈P Q, v→ 〉 < ||P Q||. first move. The first player unable to Thus, by switching the partners of R and l  1 move loses the game. Who wins this R in M one could decrease the total length 2 We now fix some arbitrary point Q in the game, if both players play optimally? of M. Therefore, M is crossing-free. plane, and we define the Q-cost of assigning We will show that Player 1 has a winning → Hershberger & Suri [8] construct → wedge W to point P as 〈P Q, v 〉. We strategy if and only if G has no perfect crossing-free matchings in O(n log n) time. k i i k compute a matching M between wedges matching. It is easy to see that Player 2 has a Their approach is purely based on geometric and points that maximizes the total Q-cost. winning strategy, if G does contain a perfect observations, and does not use graph theory. By renumbering the points, we may assume matching M: Whenever Player 1 moves to Their time complexity O(n log n) is best that for k = 1, . . . , n matching M assigns some vertex v, then Player 2 reacts by simply possible, since the problem contains sorting wedge W to point P , and that hence the moving to the partner vertex of v in M. as a special case. k k translated wedges are Wk[Pk]. Here is the And Player 1 wins the game, if G does not first beautiful observation: contain a perfect matching: Player 1 fixes 3 Illuminating the ocean an arbitrary maximum cardinality matching In the following problem, the reader should Lemma 3 If matching M maximizes M (that by assumption is non-perfect), and think of the plane as an ocean, of the points the Q-cost, then point Q is covered then starts the path in a vertex u' that is not covered by M. Whenever Player 2 moves Pk as lighthouses in this ocean, and of the by one of the translated wedges. to some vertex v, Player 1 reacts by moving wedges Wk as regions that are illuminated to the partner vertex of v in M. Note that by floodlights: There are n points P1, . . . Proof. Suppose not. Observe that for every Player 2 must always move to a vertex , Pn in the Euclidean plane. Furthermore, point Pi, there exists some wedge Wk with Q u'' that is covered by M; otherwise, the there are n rays that emanate from the ∈ Wk[Pi]; we denote this situation by origin and cut the Euclidean plane into n i → k. Clearly, the relation → contains some augmenting path P from u' to u'' could be used to increase the cardinality of M. wedges W1, . . . ,Wn. These wedges span directed cycle c1 → c2 → · · · → cs → cs+1 n This game shows up (for instance) angles α1, . . . , αn with ∑ k=1 αk = 2π. = c1 for some s ≥ 2. But then the following Show: The wedges W , . . . , W can be cyclic switch would increase the Q-cost of as an exercise in Bondy & Murty [2]. 1 n Interestingly, Bodlaender [1] has proved that translated from the origin to the points P , M: For k = 1, . . . , s re-assign wedge Wc 1 k+1 the corresponding path-forming game in . . . , Pn (one wedge per point), such that from point Pc to point Pc . By Lemma 2, k+1 k directed graphs is PSPACE-complete, and their translates again cover the entire plane. this contradicts the maximality of M.  hence intractable. The special case of this problem with n = 4

points and with angles αk ≡ π/2 was posed Here is the second beautiful observation: at the 1967 All Soviet Union Mathematical The Q-costs do depend on the choice of op t i m a 7 3 January 2007 page 

References [1] H.L. Bodlaender (1993). [4] H.N. Gabow and R.E. [6] M.R. Garey and D.S. Johnson [8] J. Hershberger and S. Suri Complexity of path-forming Tarjan (1985). A linear-time (1979). and (1992). Applications of a games. Theoretical Computer algorithm for a special case Intractability: A Guide to the semi-dynamic convex hull Science 110, 215{245. of disjoint set union. Journal Theory of NP-Completeness. algorithm. BIT 32, 249{267. of Computer and System Freeman, San Francisco. [2] J.A. Bondy and U.S.R. Murty Sciences 30, 209{221. [9] A. Schrijver (2003). (1976). Graph Theory with [7] A.M.H. Gerards (1995). Combinatorial Optimization: Applications. Elsevier/ [5] V. Galperin and G. Galperin Matching. In: Network Polyhedra and E±ciency. MacMillan, New York. (1981). Osveschenije Models, M.O. Ball, T.L. Springer, Berlin, Heidelberg. ploskosti prozhektorami Magnanti, C.L. Monma, [3] M. Fujii, T. Kasami, and K. (Illuminating a plane with and G.L. Nemhauser (eds.), Ninomiya (1969). Optimal spotlights). Kvant 11, 28{30. Handbooks in Operations sequencing of two equivalent (In Russian). Research and Management processors. SIAM Journal Science, Volume 7, 135{224. of Applied Mathematics 17, 784{789.

Symposium in honor of George Nemhauser A two-day symposium will be held in honor cover George's work and his contributions The Symposium will be held at Georgia of George's 70th birthday. George has had through various stages of his career. Tech, start at noon on Thursday, a profound impact on many people in Speakers at the symposium include: July 26, 2007 and end at 1:30 on the operations research and mathematical Cindy Barnhart Tom Magnanti Friday, July 27 (George's birthday is programming community. He was Bob Bixby Bill Pulleyblank the 27th). There will be a banquet on president of MPS from 1989-1992. This Bill Cook Don Ratliff Thursday evening. The registration event will provide the participants with an Gerard Cornuejols David Ryan fee is $100. Further announcements opportunity to show their appreciation. The Marshall Fisher Mike Todd and registration details will follow. symposium will include a small number Rob Garfinkel Mike Trick Mike Ball & Martin Savelsbergh of scientific presentations on general Martin Groetschel Laurence Wolsey topics and also special presentations that John Jarvis

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Peter Ladislaw Hammer December 23, 1936 – December 27, 2006

Peter Ladislaw Hammer was born in publications include 19 books and over 240 of Boolean functions or Boolean equations Timisoara, , on December 23, scientific papers. (See the Web site rutcor. to investigate the underlying structure, 1936. He earned his Ph.D. in mathematics rutgers.edu for a complete bibliography.) the “essence” of the problems themselves. under Academician Grigore C. Moisil at At the very onset of his career, as a More than often, this goal is met through the University of in 1966. He researcher at the Institute of Mathematics a simplifying process based, once again, on defected to Israel in 1967 where he became of the Academia of Romania, Peter the tools of . This approach a professor at the Technion in Haifa. After Hammer wrote several important articles provides, for instance, a simple way to moving to Canada, he taught from 1969 on transportation problems, jointly with demonstrate that every system of linear to 1972 at the University of Montreal, Egon Balas. At the same time his advisor, inequalities in binary variables is equivalent and from 1972 to 1983 at the University Grigore Moisil, directed him to the study to a set of inequalities involving only 0,1,-1 of Waterloo. In 1983, he moved to the of Boolean algebra. In this field, a central coefficients, as observed in a joint paper by USA and became a professor at Rutgers role is played by functions depending on Frieda Granot and Peter Hammer (1972). It University, where he founded RUTCOR binary variables, and taking either binary also led Peter Hammer, Ellis Johnson and - the Rutgers Center for Operations values (i.e., Boolean functions) or real values Uri Peled (1975) to early investigations into Research. He remained the director of (i.e., pseudo-Boolean functions). In a series the facial structure of knapsack polyhedra. RUTCOR until his untimely death in a of papers, Peter Hammer demonstrated In a related stream of research, Peter that a large variety of relevant problems Hammer has established numerous fruitful tragic car accident, on December 27, 2006. of operations research, combinatorics and links between graph theory and Boolean For more than 40 years, Peter Hammer computer science can be reduced to the functions. In a famous joint paper with has ranked among the most influential optimization of a pseudo-Boolean function VaŠek Chvátal on the aggregation of researchers in the fields of operations under constraints described by a system inequalities in integer programming (1977), research and discrete mathematics. He of pseudo-Boolean inequalities. A further he introduced and characterized the class has made numerous major contributions main conceptual step in his work was of threshold graphs, inspired by threshold to these fields, launching several new the characterization of the set of feasible Boolean functions. Threshold graphs have research directions. His results have solutions of the above system as solutions of subsequently been the subject of scores of influenced hundreds of colleagues and a single Boolean equation (or, equivalently, articles and of a book by Mahadev and have made a lasting impact on many areas of a satisfiability problem). This led him, in Uri Peled, two of Peter Hammer’s former of mathematics, computer science, and joint work with Ivo Rosenberg and Sergiu doctoral students. Other links between statistics. Rudeanu, to the development of an original graphs and Boolean or pseudo-Boolean Most of Peter Hammer’s scientific approach inspired from classical Boolean functions have been explored in joint work production has its roots in the work of methods for the solution of a large variety of with Claude Benzaken, Dominique de George Boole on propositional . More discrete optimization problems. Werra, Stephan Foldes, Toshihide Ibaraki, than anyone else, Peter Hammer has used This research project culminated in Alex Kelmans, Vadim Lozin, Frédéric and extended Boole’s machina universalis 1968 with the publication of the book Maffray, Bruno Simeone, etc. to handle questions relating to decision Boolean Methods in Operations Research Quadratic 0-1 optimization has been making, analysis and synthesis as they arise and Related Areas (Springer-Verlag, 1968), one of Peter Hammer’s main fields of in natural, economic and social sciences. co-authored by Sergiu Rudeanu. This investigation. The theory of roof-duality Over the span of his scientific career, he landmark monograph, which founded the (1984), jointly developed with Pierre has conducted eclectic forays into the field of pseudo-Boolean optimization, has Hansen and Bruno Simeone, builds on interactions between Boolean methods, influenced several generations of students concepts from linear programming (linear optimization, and combinatorial analysis, and researchers, and is now considered a relaxations), Boolean theory (quadratic while adapting his investigations to the most “classic” in Operations Research. Boolean equations) and networks recent advances of mathematical knowledge In a sense, Peter Hammer’s early (maximum network flow problems) to and of various fields of application. work can be viewed as a forerunner of compute best linear approximations of Among the main research topics which subsequent developments in the theory quadratic pseudo-Boolean functions and have received his attention, one finds an of computational complexity, since it tight bounds on the maximum value of such impressive array of methodological studies was in effect demonstrating that a large functions. Further research along similar dealing with combinatorial optimization, class of combinatorial optimization lines has been conducted by Peter Hammer some excursions into logistics and game problems is reducible to the solution of in collaboration with Endre Boros, Jean- theory, numerous contributions to graph Boolean equations. However, this purely Marie Bourjolly, Yves Crama, David Rader, theory, to the algorithmic aspects of “reductionist” view of his work would Gabriel Tavares, X. Sun, etc. propositional logic, to artificial intelligence be quite narrow. In fact, Peter Hammer Peter Hammer has also shown interest and, more recently, to the development of has systematically used the “canonical” for the application of Boolean models in innovative data mining techniques. His representation of various problems in terms op t i m a 7 3 January 2007 page 10

artificial intelligence and related fields, as and editor-in-chief of several highly- Honoring Peter L. Hammer (Caesarea witnessed by numerous joint papers with rated professional journals, including Rothchild Institute, University of Haifa, Gabriela and Sorin Alexe, Martin Anthony, Discrete Mathematics, Discrete Applied 2003), and the International Conference Tiberius Bonates, Endre Boros, Yves Crama, Mathematics, Discrete Optimization, on Graphs and Optimization (GO V, Oya Ekin, Toshi Ibaraki, Alex Kogan, Annals of Discrete Mathematics, Annals Leukerbad, Switzerland, 2006). Miguel Lejeune, Irina Lozina, and other of Operations Research and the SIAM Peter Hammer was not only an collaborators. His contributions bear on Monographs on Discrete Mathematics and outstanding scholar and a tireless organizer, automatic theorem proving, compression Applications. At Rutgers University, Peter but also a kind, generous and humorous of knowledge bases, algorithms for special Hammer was the founding Director of the human being. He relished the interaction classes of satisfiability problems, etc. About operations research programme, and he was with students and colleagues, and made 20 years ago, he launched an innovative largely responsible for developing RUTCOR everybody feel comfortable to work with approach to data mining based on a blend into an internationally recognized center him, be it on a mathematical question of Boolean techniques and combinatorial of excellence and an open institute, where (which he was always keen to formulate) or optimization. The basic tenets of this seminars, workshops, graduate courses, and on planning a conference. He supervised approach were presented in a joint paper a constant flow of visitors create a buzzing numerous graduate students with respect with Yves Crama and Toshihide Ibaraki and stimulating research environment. He and fatherly understanding, considering (1988) and were subsequently developed was also a tireless organizer of professional each one of them as his “best student”. He by Peter Hammer and his coworkers into conferences and workshops, where he always was also a true “citizen of the world”: born a new broad area of research, which he made sure to provide opportunities for in Romania from a Hungarian family, he dubbed Logical Analysis of Data, or LAD interactions between experienced scientists subsequently took the Canadian citizenship, for short. The effectiveness of the LAD and younger researchers. then the US one, wrote joint papers with methodology has been validated by many The importance of Peter Hammer’s co-authors of 28 different countries, fluently successful applications to real-life data scientific contribution was acknowledged spoke 6 languages (or more), travelled the analysis problems. In particular, some front- by the award of numerous international world extensively, spent extended periods of of-the line medical centers are increasingly distinctions, including the “George time in Belgium, France, Israel, Italy, Russia, using LAD in the actual practice of medical Tzitzeica” prize of the Switzerland and many other countries, and diagnosis for a variety of syndromes. of Science (1966), the Euler Medal of developed an extended network of friends Many aspects of Peter Hammer’s the Institute of Combinatorics and and coworkers on all continents. immense contribution to the study of its Applications (1999), and honorary Finally, last but certainly not least, Peter Boolean functions and their combinatorial degrees from the Swiss Federal Institute Hammer was a loving husband, father and structure are to be found in a forthcoming of Technology in Lausanne (1986), the grandfather. He is survived by his wife, monograph entitled Boolean Functions: University of Rome “La Sapienza” (1998), Anca Ivanescu, whom he married in 1961 Theory, Algorithms, and Applications, co- and the University of Liège (1999). He was and whose family name he assumed for a authored by Yves Crama and several other a Fellow of the American Association for few years, by his two sons Alexander and close collaborators of Peter Hammer, to be the Advancement of Science since 1974, Maxim, and by four beloved grandchildren, published by Cambridge University Press in and a Founding Fellow of the Institute Isabelle, Madeline, Annelise, and Oliver. 2007. of Combinatorics and its Applications. He will be missed by everyone who knew Beside his scientific production, Peter Several conferences were organized in his him, always and forever. Hammer will undoubtedly be remembered honor, including the First International for his vigorous contribution to and Colloquium on Pseudo-Boolean Endre Boros1, Yves Crama2 promotion of discrete mathematics and Optimization (Chexbres, Switzerland, and Bruno Simeone3 operations research. He was the founder 1987), the Workshop and Symposia

1 RUTCOR, Rutgers Center for Operations Research, Piscataway, NJ 08854-8003, USA 2 HEC Management School, University of Liège, 4000 Liège, Belgium 3 Department of Statistics, Probability and Applied Statistics, University La Sapienza, 00185 Rome, Italy op t i m a 7 3 January 2007 page 11 Call for Manuscripts MPS-SIAM Book Series on OPTIMIZATION 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 YOU ARE quality. INVITED TO CONTRIBUTE BOOKS IN THE SERIES INCLUDE: If you are interested in submitting a proposal Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization or manuscript for Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille publication in the 2005 · xii + 634 pages · Softcover series or would like 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 additional information, please contact: Applications of Stochastic Programming Edited by Stein W.Wallace and William T. Ziemba Michael L. Overton 2005 · xvi + 709 pages · Softcover Courant Institute of ISBN-13: 978-0-898715-55-2 · ISBN-10: 0-89871-555-5 Mathematical Sciences List Price $142.00 · MPS/SIAM Member Price $99.40 · Order Code MP05 New York University The Sharpest Cut: [email protected] The Impact of Manfred Padberg and His Work OR Edited by Martin Grötschel Sara J. Murphy 2004 · xi + 380 pages · Hardcover Series Acquisitions Editor ISBN-13: 978-0-898715-52-1 · ISBN-10: 0-89871-552-0 SIAM List Price 99.00 · MPS/SIAM Member Price $69.30 · Order Code MP04 [email protected] A Mathematical View of Interior-Point Methods in Convex Optimization Submission procedures, James Renegar a comprehensive 2001 · viii + 117 pages · Softcover brochure detailing 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 SIAM’s book publishing Lectures on Modern Convex Optimization: process, and other vital Analysis,Algorithms, and Engineering Applications information are Aharon Ben-Tal and Arkadi Nemirovski available from SIAM. 2001 · xvi + 488 pages · Softcover ISBN-13: 978-0-898714-91-3 · ISBN-10: 0-89871-491-5 SIAM publishes quality List Price $113.50 · MPS/SIAM Member Price $79.45 · Order Code MP02 books with practical Trust-Region Methods implementation at A. R. Conn, N. I. M. Gould, and Ph. L.Toint prices affordable to 2000 · xx + 959 pages · Hardcover ISBN-13: 978-0-898714-60-9 · ISBN-10: 0-89871-460-5 individuals. List Price $136.50 · MPS/SIAM Member Price $95.55 · Order Code MP01

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