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54
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 704 (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.
Probabilistic checking of proofs: a new characterization of NP
 Journal of the ACM
, 1998
"... Abstract. 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 ..."
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Cited by 357 (28 self)
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Abstract. 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 SubConstant ErrorProbability LowDegree Test, and a SubConstant ErrorProbability PCP Characterization of NP
 IN PROC. 29TH ACM SYMP. ON THEORY OF COMPUTING, 475484. EL PASO
, 1997
"... We introduce a new lowdegreetest, one that uses the restriction of lowdegree polynomials to planes (i.e., affine subspaces of dimension 2), rather than the restriction to lines (i.e., affine subspaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, ..."
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Cited by 274 (19 self)
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We introduce a new lowdegreetest, one that uses the restriction of lowdegree polynomials to planes (i.e., affine subspaces of dimension 2), rather than the restriction to lines (i.e., affine subspaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, much smaller than constant). The new test enables us to prove a lowerror characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ffl ? 0, membership in any NP language can be verified with O(1) accesses, each reading logarithmic number of bits, and such that the errorprobability is 2 \Gamma log 1\Gammaffl n . Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SETCOVER to within a logarithmic factors is NPhard. Previous analysis for lowdegreetests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error...
Improved Decoding of ReedSolomon and AlgebraicGeometry Codes
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1999
"... Given an errorcorrecting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding ReedSolomon codes ..."
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Cited by 244 (40 self)
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Given an errorcorrecting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding ReedSolomon codes. The list decoding problem for ReedSolomon codes reduces to the following "curvefitting" problem over a field F : Given n points f(x i :y i )g i=1 , x i
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 101 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
Some Applications of Coding Theory in Computational Complexity
, 2004
"... Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory ..."
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Cited by 49 (2 self)
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Errorcorrecting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locallytestable and locallydecodable errorcorrecting codes, and their applications to complexity theory and to cryptography.
Towards 3query locally decodable codes of subexponential length
 In Proc. of the 39th ACM Symposium on Theory of Computing (STOC
, 2007
"... A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new construc ..."
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Cited by 47 (5 self)
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A qquery Locally Decodable Code (LDC) encodes an nbit message x as an Nbit codeword C(x), such that one can probabilistically recover any bit xi of the message by querying only q bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. We give new constructions of three query LDCs of vastly shorter length than that of previous constructions. Specifically, given any Mersenne prime p = 2 t − 1, we design three query LDCs of length N = exp � n 1/t �, for every n. Based on the largest known Mersenne prime, this translates to a length of less than exp n 10−7� compared to exp � n1/2 � in the previous constructions. It has often been conjectured that there are infinitely many Mersenne�primes. Under this conjecture, our constructions yield three query locally decodable codes of length N = exp n O
Some Improvements to Total Degree Tests
, 1995
"... A lowdegree test is a collection of simple, local rules for checking the proximity of an arbitrary function to a lowdegree polynomial. Each rule depends on the function’s values at a small number of places. If a function satisfies many rules then it is close to a lowdegree polynomial. Lowdegree ..."
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Cited by 43 (9 self)
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A lowdegree test is a collection of simple, local rules for checking the proximity of an arbitrary function to a lowdegree polynomial. Each rule depends on the function’s values at a small number of places. If a function satisfies many rules then it is close to a lowdegree polynomial. Lowdegree tests play an important role in the development of probabilistically checkable proofs. In this paper we present two improvements to the efficiency of lowdegree tests. Our first improvement concerns the smallest field size over which a lowdegree test can work. We show how to test that a function is a degree d polynomial over prime fields of size only d + 2. Our second improvement shows a better efficiency of the lowdegree test of [ 141 than previously known. We show concrete applications of this improvement via the notion of “locally checkable codes”. This improvement translates into better tradeoffs on the size versus probe complexity of probabilistically checkable proofs than previously known.
Random Debaters and the Hardness of Approximating Stochastic Functions
, 1994
"... . A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomialtime verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V ..."
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Cited by 29 (6 self)
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. A probabilistically checkable debate system (PCDS) for a language L consists of a probabilistic polynomialtime verifier V and a debate between Player 1, who claims that the input x is in L, and Player 0, who claims that the input x is not in L. It is known that there is a PCDS for L in which V flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE ([Condon et al., Proc. 25th ACM Symposium on Theory of Computing, 1993, pp. 304315]). In this paper, we restrict attention to RPCDS's, which are PCDS's in which Player 0 follows a very simple strategy: On each turn, Player 0 chooses uniformly at random from the set of legal moves. We prove the following result. Theorem: L has an RPCDS in which the verifier flips O(logn) coins and reads O(1) bits of the debate if and only if L is in PSPACE. This new characterization of PSPACE is used to show that certain stochastic PSPACEhard functions are as hard to approximate closely as they are to compute exactly. Exam...