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295
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 974 (14 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 723 (45 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 Euclidean TSP and other geometric problems
 In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96
, 1996
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes a ..."
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Cited by 336 (3 self)
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Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
Free Bits, PCPs and NonApproximability  Towards Tight Results
, 1996
"... This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems. ..."
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Cited by 203 (41 self)
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This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.
Multipoint Relaying: An Efficient Technique for flooding in Mobile Wireless Networks
, 2000
"... In this paper we discuss the mechanism of multipoint relays (MPRs) to efficiently do the ooding of broadcast messages in the mobile wireless networks. Multipoint relaying is a technique to reduce the number of redundant retransmissions while diffusing a broadcast message in the network. We discuss t ..."
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Cited by 183 (6 self)
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In this paper we discuss the mechanism of multipoint relays (MPRs) to efficiently do the ooding of broadcast messages in the mobile wireless networks. Multipoint relaying is a technique to reduce the number of redundant retransmissions while diffusing a broadcast message in the network. We discuss the principle and the functioning of MPRs, and propose a heuristic to select these MPRs in a mobile wireless environment. We also analyze the complexity of this heuristic and prove that the computation of a multipoint relay set with minimal size is NPcomplete. Finally, we present some simulation results to show the efficiency of multipoint relays.
Complexity and Approximation
, 1999
"... Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance ..."
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Cited by 180 (1 self)
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Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance I. How can we exploit the knowledge we have on the solution of instance I to compute a (approximate) solution of instance I ′ in an efficient way? This computation model is called reoptimization and is of practical interest in various circumstances. In this article we first discuss what kind of performance we can expect for specific classes of problems and then we present some classical optimization problems (i.e. Max Knapsack, Min Steiner Tree, Scheduling) in which this approach has been fruitfully applied. Subsequently, we address vehicle routing problems and we show how the reoptimization approach can be used to obtain good approximate solution in an efficient way for some of these problems. 1
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 178 (28 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
Multipoint relaying for flooding broadcast messages in mobile wireless networks
, 2002
"... In this paper we discuss the mechanism of multipoint relays (MPRs) to efficiently do the flooding of broadcast messages in the mobile wireless networks. Multipoint relaying is a technique to reduce the number of redundant retransmissions while diffusing a broadcast message in the network. We discus ..."
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Cited by 168 (1 self)
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In this paper we discuss the mechanism of multipoint relays (MPRs) to efficiently do the flooding of broadcast messages in the mobile wireless networks. Multipoint relaying is a technique to reduce the number of redundant retransmissions while diffusing a broadcast message in the network. We discuss the principle and the functioning of MPRs, and propose a heuristic to select these MPRs in a mobile wireless environment. We also analyze the complexity of this heuristic and prove that the computation of a multipoint relay set with minimal size is NPcomplete. Finally, we present some simulation results to show the efficiency of multipoint relays.
The Quadratic Assignment Problem
 HANDBOOK OF COMBINATORIAL OPTIMIZATION, P. PARDALOS AND D.Z. DU, EDS.
, 1998
"... This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, an ..."
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Cited by 115 (3 self)
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This paper aims at describing the state of the art on quadratic assignment problems (QAPs). It discusses the most important developments in all aspects of the QAP such as linearizations, QAP polyhedra, algorithms to solve the problem to optimality, heuristics, polynomially solvable special cases, and asymptotic behavior. Moreover, it also considers problems related to the QAP, e.g. the biquadratic assignment problem, and discusses the relationship between the QAP and other well known combinatorial optimization problems, e.g. the traveling salesman problem, the graph partitioning problem, etc. The paper will appear in the Handbook of Combinatorial Optimization to be published by Kluwer Academic Publishers, P. Pardalos and D.Z. Du, eds.
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 104 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.