Results 1  10
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105
Counterexampleguided Abstraction Refinement
, 2000
"... We present an automatic iterative abstractionrefinement methodology in which the initial abstract model is generated by an automatic analysis of the control structures in the program to be verified. Abstract models may admit erroneous (or "spurious") counterexamples. We devise new symb ..."
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Cited by 848 (71 self)
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We present an automatic iterative abstractionrefinement methodology in which the initial abstract model is generated by an automatic analysis of the control structures in the program to be verified. Abstract models may admit erroneous (or "spurious") counterexamples. We devise new symbolic techniques which analyze such counterexamples and refine the abstract model correspondingly.
Some optimal inapproximability results
, 2002
"... We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for ..."
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Cited by 782 (12 self)
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We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for MaxE2Sat, MaxCut, MaxdiCut, and Vertex cover. Warning: Essentially this paper has been published in JACM and is subject to copyright restrictions. In particular it is for personal use only.
The Maximum Clique Problem
, 1999
"... Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computation ..."
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Cited by 195 (21 self)
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Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .
The PCP theorem by gap amplification
 In Proceedings of the ThirtyEighth Annual ACM Symposium on Theory of Computing
, 2006
"... The PCP theorem [3, 2] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to [12], has placed the PC ..."
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Cited by 169 (9 self)
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The PCP theorem [3, 2] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to [12], has placed the PCP theorem at the heart of the area of inapproximability. In this work we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NPhardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem. Our approach is to consider the unsat value of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsatvalue of a constraintsystem, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabetsize that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem. The amplification lemma relies on a new notion of “graph powering ” that can be applied to systems of binary constraints. This powering amplifies the unsatvalue of a constraint system provided that the underlying graph structure is an expander. We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locallytestable codes whose length is linear up to a polylog factor, and whose correctness can be probabilistically verified by making a constant number of queries. Namely, we prove SAT ∈
Vertex Cover Might be Hard to Approximate to within 2ɛ
 IN PROCEEDINGS OF THE 18TH ANNUAL IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2003
"... We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k ..."
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Cited by 150 (11 self)
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We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2prover1round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraphs is hard to approximate within any constant factor better than k.
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...
On the Hardness of Approximating Multicut and SparsestCut
 In Proceedings of the 20th Annual IEEE Conference on Computational Complexity
, 2005
"... We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1. ..."
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Cited by 102 (5 self)
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We show that the MULTICUT, SPARSESTCUT, and MIN2CNF ≡ DELETION problems are NPhard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.
Approximation Algorithms for Constraint Satisfaction Problems Involving at Most Three Variables per Constraint
 In Proceedings of the 9th Annual ACMSIAM Symposium on Discrete Algorithms
, 1997
"... An instance of MAX 3CSP is a collection of m clauses of the form f i (z i1 ; z i2 ; z i3 ), where the z ij 's are literals, or constants, from the set f0; 1; x 1 ; : : : ; xn ; x 1 ; : : : ; xng, and the f i 's are arbitrary Boolean functions depending on (at most) three variables. Eac ..."
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Cited by 92 (6 self)
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An instance of MAX 3CSP is a collection of m clauses of the form f i (z i1 ; z i2 ; z i3 ), where the z ij 's are literals, or constants, from the set f0; 1; x 1 ; : : : ; xn ; x 1 ; : : : ; xng, and the f i 's are arbitrary Boolean functions depending on (at most) three variables. Each clause has a nonnegative weight w i associated with it. A solution to the instance is an assignment of 01 values to the variables x 1 ; : : : ; xn that maximizes P n i=1 w i f i (z i1 ; z i2 ; z i3 ), the total weight of the satisfied clauses. The MAX 3CSP problem is clearly a generalization of the MAX 3SAT problem. (In an instance of the MAX 3SAT problem f i (z i1 ; z i2 ; z i3 ) = z i1 z i2 z i3 for every i.) Karloff and Zwick have recently obtained a 8 approximation algorithm for MAX 3SAT. Their algorithm is based on a new semidefinite relaxation of the problem. Hastad showed that no polynomial time algorithm can achieve a better performance ratio, unless P=NP. Here we use similar techniques to obtain a approximation algorithm for MAX 3CSP. The performance ratio of this algorithm is also optimal, as follows again from the work of Hastad. We also obtain better performance ratios for several special cases of the problem. Our results include: 2 approximation algorithm for MAX 3AND, the problem in which each clause is of the form z i1 z i2 z i3 . This result is optimal and it implies the result for MAX 3CSP.
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1 ..."
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Cited by 90 (8 self)
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1