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137
PseudoBoolean Optimization
 DISCRETE APPLIED MATHEMATICS
, 2001
"... This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of ..."
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Cited by 183 (5 self)
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This survey examines the state of the art of a variety of problems related to pseudoBoolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudoBoolean optimization, the specially important case of quadratic pseudoBoolean optimization (to which every pseudoBoolean optimization can be reduced), several other important special classes, and approximation algorithms.
Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT
 IN PROCEEDINGS OF THE THIRD ISRAEL SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS
, 1995
"... It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the FeageLovdsz ..."
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Cited by 142 (10 self)
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It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the FeageLovdsz (STOC92) semidefinite programming relaxation of oneround twoprover proof systems, together with rounding techniques for the solutions of semidefinite progmms, as introduced by Goemans and Williamson (STO C94). As a consequence of our approach, we present improved approximation algorithms for MAX 2SAT and MAX DICUT. The algorithms are guamnteed to deliver solutions within a factor of 0.931 of the optimum for MAX 2SAT and within a factor of 0.859 for MAX DICUT, improving upon the guarantees of 0.878 and 0.796 of Goemans and Williamson.
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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A 7/8Approximation Algorithm for MAX 3SAT?
 IN PROCEEDINGS OF THE 38TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1997
"... We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instancea collection of clauses each of length at most threeis satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a p ..."
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Cited by 112 (10 self)
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We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instancea collection of clauses each of length at most threeis satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAXCUT algorithm of Goemans and Williamson and the MAX 2SAT algorithm of Feige and Goemans. Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P != NP , no polynomialtime algorithm for MAX 3SAT can achieve a performance ratio exceeding 7=8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semi...
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.
New outer bounds on the marginal polytope
 In Advances in Neural Information Processing Systems
, 2007
"... We give a new class of outer bounds on the marginal polytope, and propose a cuttingplane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Mar ..."
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Cited by 74 (6 self)
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We give a new class of outer bounds on the marginal polytope, and propose a cuttingplane algorithm for efficiently optimizing over these constraints. When combined with a concave upper bound on the entropy, this gives a new variational inference algorithm for probabilistic inference in discrete Markov Random Fields (MRFs). Valid constraints on the marginal polytope are derived through a series of projections onto the cut polytope. As a result, we obtain tighter upper bounds on the logpartition function. We also show empirically that the approximations of the marginals are significantly more accurate when using the tighter outer bounds. Finally, we demonstrate the advantage of the new constraints for finding the MAP assignment in protein structure prediction. 1
On the optimality of the random hyperplane rounding technique for MAX CUT
 Algorithms
, 2000
"... MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NPhard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the grap ..."
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Cited by 57 (3 self)
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MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NPhard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least ff \Delta sdp, where ff ' 0:87856 and sdp is the value of the semidefinite program, which is an upper bound on opt, the number of edges in the maximum cut. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is ff. The previously known bound on the integrality ratio was roughly 0:8845. 2. In the presence of the so called "triangle constraints", the integrality ratio is no better than roughly 0:891. The previously known bound was above ...
Eigenvalues in combinatorial optimization
, 1993
"... In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey. ..."
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Cited by 44 (0 self)
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In the last decade many important applications of eigenvalues and eigenvectors of graphs in combinatorial optimization were discovered. The number and importance of these results is so fascinating that it makes sense to present this survey.