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111
Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming
 Journal of the ACM
, 1995
"... We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution ..."
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Cited by 1211 (13 self)
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We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...
Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 797 (39 self)
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We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNPhard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an Nvertex graph to within a factor of N ɛ is NPhard.
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiabi ..."
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Cited by 189 (35 self)
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We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By dense graphs we mean graphs with minimum degree &Omega;(n), although our algorithms solve most of these problems so long as the average degree is &Omega;(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.
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 179 (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.
Improvements To Propositional Satisfiability Search Algorithms
, 1995
"... ... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable ..."
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Cited by 174 (0 self)
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... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable random 3SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NPcomplete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.
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 141 (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.
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 117 (5 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
Improved NonApproximability Results
, 1994
"... We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in... ..."
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Cited by 110 (13 self)
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We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in...
The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations
 Theoretical Computer Science
, 1993
"... We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of rela ..."
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Cited by 92 (11 self)
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We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, , ? and 6=. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, or ? relations is NPhard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for 6= relations with real coefficients. The various NPhard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the ran...