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Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 109 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 89 (3 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
A spectral algorithm for seriation and the consecutive ones problem
 SIAM Journal on Computing
, 1998
"... Abstract. In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire ..."
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Cited by 58 (0 self)
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Abstract. In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire for each pair of elements to be near each other, find all permutations π with the property that if π(i) < π(j) < π(k) then f(i, j) ≥ f(i, k) and f(j, k) ≥ f(i, k). This seriation problem is a generalization of the wellstudied consecutive ones problem. We present a spectral algorithm for this problem that has a number of interesting features. Whereas most previous applications of spectral techniques provide only bounds or heuristics, our result is an algorithm that correctly solves a nontrivial combinatorial problem. In addition, spectral methods are being successfully applied as heuristics to a variety of sequencing problems, and our result helps explain and justify these applications.
Cones Of Matrices And Successive Convex Relaxations Of Nonconvex Sets
, 2000
"... . Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each ..."
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Cited by 50 (19 self)
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. Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding LovaszSchrijver liftandproject procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
, 1998
"... . We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than th ..."
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Cited by 49 (11 self)
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. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different tradeoffs have been proposed. The adjacencylist encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted nnode tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...
Toward a general theory of quantum games
 In Proceedings of 39th ACM STOC
, 2006
"... Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that g ..."
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Abstract We study properties of quantum strategies, which are complete specifications of a givenparty's actions in any multipleround interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantumstrategies that generalizes the ChoiJamiol/kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting onlyon the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simpleproof of Kitaev's lower bound for strong coinflipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games. 1 Introduction The theory of games provides a general structure within which both cooperation and competitionamong independent entities may be modeled, and provides powerful tools for analyzing these models. Applications of this theory have fundamental importance in many areas of science.This paper considers games in which the players may exchange and process quantum information. We focus on competitive games, and within this context the types of games we consider arevery general. For instance, they allow multiple rounds of interaction among the players involved, and place no restrictions on players ' strategies beyond those imposed by the theory of quantuminformation. While classical games can be viewed as a special case of quantum games, it is important tostress that there are fundamental differences between general quantum games and classical games. For example, the two most standard representations of classical games, namely the normal formand extensive form representations, are not directly applicable to general quantum games. This is due to the nature of quantum information, which admits a continuum of pure (meaning extremal)
Solving project scheduling problems by minimum cut computations
 Management Science
, 2003
"... This paper also can be downloaded without charge from the ..."
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Cited by 39 (2 self)
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This paper also can be downloaded without charge from the
Derandomizing approximation algorithms based on semidefinite programming
 SIAM Journal on Computing
, 1999
"... Abstract. Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NPhard problems, namely MaxCut, Max kCut, MaxSat, MaxDicut, Maxbisection, kvertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomize ..."
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Cited by 32 (0 self)
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Abstract. Remarkable breakthroughs have been made recently in obtaining approximate solutions to some fundamental NPhard problems, namely MaxCut, Max kCut, MaxSat, MaxDicut, Maxbisection, kvertex coloring, maximum independent set, etc. All these breakthroughs involve polynomial time randomized algorithms based upon semidefinite programming, a technique pioneered by Goemans and Williamson. In this paper, we give techniques to derandomize the above class of randomized algorithms, thus obtaining polynomial time deterministic algorithms with the same approximation ratios for the above problems. At the heart of our technique is the use of spherical symmetry to convert a nested sequence of n integrations, which cannot be approximated sufficiently well in polynomial time, to a nested sequence of just a constant number of integrations, which can be approximated sufficiently well in polynomial time. Key words. NPhard, approximation algorithm, derandomization, semidefinite programming AMS subject classification. 68Q25 PII. S0097539796309326
Polynomial methods for separable convex optimization in unimodular linear spaces with applications
 SIAM J. Comput
, 1997
"... We consider the problem of minimizing a separable convex objective function over the linear space given by system Mx = 0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network, and the problems of determi ..."
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Cited by 31 (5 self)
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We consider the problem of minimizing a separable convex objective function over the linear space given by system Mx = 0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network, and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [5] and extended to some other problems [2, 4, 12, 7] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient Cancel and Tighten method. We then consider objective functions that are piecewise linear, or pure and piecewise quadratic, or piecewise mixed linear and quadratic, and show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles. Key words. separable convex optimization, unimodular linear spaces, min mean canceling, network flows
Scheduling Multiple VariableSpeed Machines
 OPERATIONS RESEARCH
, 1994
"... We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear but also incre ..."
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Cited by 19 (0 self)
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We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear but also increases throughput. We discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases. Some cases are inherently difficult so for these we give heuristics. Our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems. We create heuristics that optimally schedule a large portion of the jobs and then attempt to fit in the remainder. This also gives a method for quickly finding valid inequalities violated by the linear relaxation solution. For the problem of minimizing the sum of makespan and production costs, a rounding heuristic is within a constant factor of optimal. Our heuristics ar...