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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 822 (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.
Closest Point Search in Lattices
 IEEE TRANS. INFORM. THEORY
, 2000
"... In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, ba ..."
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Cited by 324 (2 self)
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In this semitutorial paper, a comprehensive survey of closestpoint search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closestpoint search algorithm, based on the SchnorrEuchner variation of the Pohst method, is implemented. Given an arbitrary point x 2 R m and a generator matrix for a lattice , the algorithm computes the point of that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent ViterboBoutros decoder. The improvement increases with the dimension of the lattice. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, compu...
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 224 (39 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.
Convexity, Classification, and Risk Bounds
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2003
"... Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 01 loss function. The convexity makes these algorithms computationally efficien ..."
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Cited by 167 (15 self)
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Many of the classification algorithms developed in the machine learning literature, including the support vector machine and boosting, can be viewed as minimum contrast methods that minimize a convex surrogate of the 01 loss function. The convexity makes these algorithms computationally efficient. The use of a surrogate, however, has statistical consequences that must be balanced against the computational virtues of convexity. To study these issues, we provide a general quantitative relationship between the risk as assessed using the 01 loss and the risk as assessed using any nonnegative surrogate loss function. We show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function: that it satisfy a pointwise form of Fisher consistency for classification. The relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications. We also present a refined version of this result in the case of low noise. Finally, we
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.
On the Approximability of Minimizing Nonzero Variables Or Unsatisfied Relations in Linear Systems
, 1997
"... We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasib ..."
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Cited by 115 (3 self)
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We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , ? and 6= are considered. While Min RVLS with equations was known to be NPhard in [27], we established in [2, 5] that Min ULR with equalities and inequalities are NPhard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in [8]. In this paper we determine strong bou...
On the Limits of NonApproximability of Lattice Problems
, 1998
"... We show simple constantround interactive proof systems for problems capturing the approximability, to within a factor of p n, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector problem (SVP). These interactive proofs are for th ..."
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Cited by 102 (3 self)
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We show simple constantround interactive proof systems for problems capturing the approximability, to within a factor of p n, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector problem (SVP). These interactive proofs are for the "coNP direction"; that is, we give an interactive protocol showing that a vector is "far" from the lattice (for CVP), and an interactive protocol showing that the shortestlatticevector is "long" (for SVP). Furthermore, these interactive proof systems are HonestVerifier Perfect ZeroKnowledge. We conclude that approximating CVP (resp., SVP) within a factor of p n is in NP " coAM. Thus, it seems unlikely that approximating these problems to within a p n factor is NPhard. Previously, for the CVP (resp., SVP) problem, Lagarias et. al., Hastad and Banaszczyk showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in NP " coNP . On the other hand, Ar...
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...
Hardness of approximating the shortest vector problem in high Lp norms
 In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science. IEEE Computer
"... Abstract. Let p> 1beany fixed real. We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓp norm within a constant factor. Under the stronger assumption NP ⊆ RTIME(2poly(log n)), we show that there is no polynomialtime (log ..."
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Cited by 91 (3 self)
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Abstract. Let p> 1beany fixed real. We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓp norm within a constant factor. Under the stronger assumption NP ⊆ RTIME(2poly(log n)), we show that there is no polynomialtime (log n)1/2−ɛ algorithm with approximation ratio 2 where n is the dimension of the lattice and ɛ>0isan arbitrarily small constant. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product,anew (log n)1/2−ɛ variant of tensor product that we introduce. This enables us to boost the hardness factor to 2.
The Approximability of Constraint Satisfaction Problems
, 2000
"... ... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstrain ..."
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Cited by 85 (1 self)
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... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstraints). MaxOnes(MinOnes)isthe classofoptimizationproblemswherethegoalistondan assignmentsatisfyingallconstraints withmaximum(minimum)numberofvariablesset to 1. Eachclassconsistsofinnitelymany thatdescribethepossibleconstraintsthatmaybeused. problemsandaproblemwithinaclass is specified by a finite collectionofniteBooleanfunctions pletelyclassiesalloptimizationproblems derived from Booleanconstraintsatisfaction.Our Creignou [11]. Inthisworkwedeterminetightboundsonthe "approximability"(i.e.,thera in MaxOnes,MinCSPandMinOnes.Combinedwiththeresultof Creignou,thiscomtiotowithinwhicheachproblemmay be approximatedinpolynomialtime)ofeveryproblem Tightboundsontheapproximabilityofeveryproblemin MaxCSPwereobtainedby resultscaptureadiversecollectionofoptimization problemssuchasMAX3SAT,MaxCut, (in)approximabilityoftheseoptimizationproblems andyieldacompactpresentationofmost MaxClique,MinCut,NearestCodewordetc. Ourresultsunifyrecentresultsonthe knownresults. Moreover, theseresultsprovideaformalbasistomanystatementsonthe behaviorofnaturaloptimizationproblems,thathaveso faronlybeenobservedempirically.