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117
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.
Probabilistic checking of proofs: a new characterization of NP
 JOURNAL OF THE ACM
, 1998
"... We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof ..."
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Cited by 414 (26 self)
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We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NPhard.
NonApproximability Results for Optimization Problems on Bounded Degree Instances
, 2001
"... We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approxima ..."
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Cited by 90 (4 self)
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We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor 7=8+O(1= B), unless RP = NP . Hastad [18] proved that the problem is approximable to within a factor 7=8+1=64B in polynomial time, and that is hard to approximate to within a factor 7=8 + 1=(log B) 3 . Our result uses a new randomized reduction from general instances of Max 3SAT to boundedoccurrences instances. The randomized reduction applies to other Max SNP problems as well.
Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems
, 1999
"... We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and William ..."
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Cited by 63 (7 self)
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We present a tool, outward rotations, for enhancing the performance of several semidefinite programming based approximation algorithms. Using outward rotations, we obtain an approximation algorithm for MAX CUT that, in many interesting cases, performs better than the algorithm of Goemans and Williamson. We also obtain an improved approximation algorithm for MAX NAEf3gSAT. Finally, we provide some evidence that outward rotations can also be used to obtain improved approximation algorithms for MAX NAESAT and MAX SAT.
Proving Integrality Gaps Without Knowing the Linear Program
 THEORY OF COMPUTING
, 2002
"... Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods see ..."
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Cited by 61 (2 self)
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Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.
Hardness of Approximating the Minimum Distance of a Linear Code
, 2003
"... We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the b ..."
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Cited by 56 (6 self)
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We show that the minimum distance d of a linear code is not approximable to within anyconstant factor in random polynomial time (RP), unless NP (nondeterministic polynomial time) equals RP. We also show that the minimum distance is not approximable to within an additiveerror that is linear in the block length n of the code. Under the stronger assumption that NPis not contained in RQP (random quasipolynomial time), we show that the minimum distance is not approximable to within the factor 2log 1ffl(n), for any ffl> 0. Our results hold for codes over any finite field, including binary codes. In the process we show that it is hard to findapproximately nearest codewords even if the number of errors exceeds the unique decoding radius d/2 by only an arbitrarily small fraction ffld. We also prove the hardness of the nearestcodeword problem for asymptotically good codes, provided the number of errors exceeds (2
Subexponential Parameterized Algorithms Collapse the Whierarchy (Extended Abstract)
, 2001
"... It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of pr ..."
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Cited by 54 (3 self)
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It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]hard.
Hardness of Approximation for VertexConnectivity NetworkDesign Problems
, 2002
"... In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths con ..."
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Cited by 50 (4 self)
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In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.
Conditional hardness for approximate coloring
 In STOC 2006
, 2006
"... We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For ..."
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Cited by 46 (14 self)
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We study the APPROXIMATECOLORING(q, Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q (where χ(G) is the chromatic number of G). We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot’s 2to1 conjecture [Khot’02]. For q = 3, we base our hardness result on a certain ‘⊲< shaped ’ variant of his conjecture. We also prove that the problem ALMOST3COLORINGε is hard for any constant ε> 0, assuming Khot’s Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε. Our result is based on bounding various generalized noisestability quantities using the invariance principle of Mossel et al [MOO’05].