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149
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.
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 776 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
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 751 (11 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.
NonDeterministic Exponential Time has TwoProver Interactive Protocols
"... We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language ..."
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Cited by 416 (37 self)
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We determine the exact power of twoprover interactive proof systems introduced by BenOr, Goldwasser, Kilian, and Wigderson (1988). In this system, two allpowerful noncommunicating provers convince a randomizing polynomial time verifier in polynomial time that the input z belongs to the language L. It was previously suspected (and proved in a relativized sense) that coNPcomplete languages do not admit such proof systems. In sharp contrast, we show that the class of languages having twoprover interactive proof systems is nondeterministic exponential time. After the recent results that all languages in PSPACE have single prover interactive proofs (Lund, Fortnow, Karloff, Nisan, and Shamir), this represents a further step demonstrating the unexpectedly immense power of randomization and interaction in efficient provability. Indeed, it follows that multiple provers with coins are strictly stronger than without, since NEXP # NP. In particular, for the first time, provably polynomial time intractable languages turn out to admit “efficient proof systems’’ since NEXP # P. We show that to prove membership in languages in EXP, the honest provers need the power of EXP only. A consequence, linking more standard concepts of structural complexity, states that if EX P has polynomial size circuits then EXP = Cg = MA. The first part of the proof of the main result extends recent techniques of polynomial extrapolation of truth values used in the single prover case. The second part is a verification scheme for multilinearity of an nvariable function held by an oracle and can be viewed as an independent result on program verification. Its proof rests on combinatorial techniques including the estimation of the expansion rate of a graph.
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.
A Parallel Repetition Theorem
 SIAM Journal on Computing
, 1998
"... We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the t ..."
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Cited by 362 (9 self)
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We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc. 11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 7076].
Algebraic Methods for Interactive Proof Systems
, 1990
"... We present a new algebraic technique for the construction of interactive proof systems. We use our technique to prove that every language in the polynomialtime hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP=PSPACE (Shamir) and that MIP ..."
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Cited by 338 (28 self)
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We present a new algebraic technique for the construction of interactive proof systems. We use our technique to prove that every language in the polynomialtime hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP=PSPACE (Shamir) and that MIP=NEXP (Babai, Fortnow and Lund).
Checking Computations in Polylogarithmic Time
, 1991
"... . Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every no ..."
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Cited by 260 (10 self)
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. Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every nondeterministic computational task S(x; y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S 0 such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in polylogarithmic Monte Carlo time; and (iii) a witness satisfying S 0 can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in errorcorrecting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N O(log log N)...
Interactive proofs and the hardness of approximating cliques
 JOURNAL OF THE ACM
, 1996
"... The contribution of this paper is twofold. First, a connection is shown between approximating the size of the largest clique in a graph and multiprover interactive proofs. Second, an efficient multiprover interactive proof for NP languages is constructed, where the verifier uses very few random b ..."
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Cited by 170 (11 self)
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The contribution of this paper is twofold. First, a connection is shown between approximating the size of the largest clique in a graph and multiprover interactive proofs. Second, an efficient multiprover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multiprover interactive proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.
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 166 (8 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 ∈