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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 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 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 324 (20 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 lowdegree testing and its applications
 IN 29TH STOC
, 1997
"... NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The stro ..."
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Cited by 142 (17 self)
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NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The strongest previously known connection for this test states that a function passes the test with probability 6 for some d> 7/8 iff the function has agreement N 6 with a polynomial of degree d. We presenta new, and surprisingly strong,analysiswhich shows thatthepreceding statementis truefor 6<<0.5. The analysis uses a version of Hilbe?l irreducibility, a tool used in the factoring of multivariate polynomials. As a consequence we obtain an alternate construction for the following proof system: A constant prover lround proof system for NP languages in which the verifier uses O(log n) random bits, receives answers of size O(log n) bits, and has an error probability of at most 2 – 10g*‘’. Such a proof system, which implies the NPhardness of approximating Set Cover to within fl(log n) factors, has already been obtained by Raz and Safra [28]. Our result was completed after we heard of their claim. A second consequence of our analysis is a self testerlcorrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on 6 fraction of inputs where 15<<0.5, then the tester/corrector determines J and generates 0(~) randomized programs, such that one of the programs is correct on every input, with high probability.
Locally Testable Codes and PCPs of AlmostLinear Length
, 2002
"... Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using ..."
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Cited by 69 (18 self)
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Locally testable codes are errorcorrecting codes that admit very efficient codeword tests. Specifically, using
SubConstant Error Low Degree Test of Almost Linear Size
 In STOC
, 2006
"... Given a function f: � m → � over a finite field �, a low degree tester tests its agreement with an mvariate polynomial of total degree at most d over �. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., line ..."
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Cited by 18 (5 self)
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Given a function f: � m → � over a finite field �, a low degree tester tests its agreement with an mvariate polynomial of total degree at most d over �. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies. We wish to minimize two parameters of a tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester’s queries. Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (P CP s) and locally testable codes (LT Cs). The error of the low degree tester is related to the soundness of the P CP and its size is related to the size of the P CP (or the length of the LT C). We design and analyze new low degree testers that have both subconstant error o(1) and almostlinear size n 1+o(1) (where n = �  m). Previous constructions of subconstant error testers had polynomial size [3, 16]. These testers enabled the construction of P CP s with subconstant soundness, but polynomial size [3, 16, 9]. Previous constructions of almostlinear size testers obtained only constant error [13, 7]. These testers were used to construct almostlinear size LT Cs and almostlinear size P CP s with constant soundness
A SubConstant ErrorProbability LowDegreeTest, and a SubConstant ErrorProbability PCP Characterization of NP
 IN PROC. 29TH ACM SYMP. ON THEORY OF COMPUTING, EL PASO
, 1997
"... We introduce a new lowdegreetest, a 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 particula ..."
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Cited by 8 (0 self)
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We introduce a new lowdegreetest, a 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 a 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 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, e...
A note on the hardness results for the labeled perfect matching problems in bipartite graphs
, 2007
"... In this note, we strengthen the inapproximation bound of O(log n) for the labeled perfect matching problem established in J. Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters 96 (2005) 8188, using a self improving operation in some hard instances. It is intere ..."
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Cited by 2 (2 self)
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In this note, we strengthen the inapproximation bound of O(log n) for the labeled perfect matching problem established in J. Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters 96 (2005) 8188, using a self improving operation in some hard instances. It is interesting to note that this self improving operation does not work for all instances. Moreover, based on this approach we deduce that the problem does not admit constant approximation algorithms for connected planar cubic bipartite graphs.
ON POLYNOMIAL SOLVABILITY OF THE HAMILTONIAN CYCLE PROBLEM FOR GRAPHS OF DEGREE LESS THAN OR EQUAL TO 3
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Electronic Colloquium on Computational Complexity, Report No. 86 (2005) SubConstant Error Low Degree Test of Almost Linear Size
, 2005
"... Given a function f: Fm → F over a finite field F, a low degree tester tests its proximity to an mvariate polynomial of total degree at most d over F. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, pl ..."
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Given a function f: Fm → F over a finite field F, a low degree tester tests its proximity to an mvariate polynomial of total degree at most d over F. The tester is usually given access to an oracle A providing the supposed restrictions of f to affine subspaces of constant dimension (e.g., lines, planes, etc.). The tester makes very few (probabilistic) queries to f and to A (say, one query to f and one query to A), and decides whether to accept or reject based on the replies. We wish to minimize two parameters of a tester: its error and its size. The error bounds the probability that the tester accepts although the function is far from a low degree polynomial. The size is the number of bits required to write the oracle replies on all possible tester’s queries. Low degree testing is a central ingredient in most constructions of probabilistically checkable proofs (P CP s) and locally testable codes (LT Cs). The error of the low degree tester is related to the soundness of the P CP and its size is related to the size of the P CP (or the length of the LT C). We design and analyze new low degree testers that have both subconstant error o(1) and almostlinear size n1+o(1) (where n = F  m). Previous constructions of subconstant error testers had polynomial size [3, 17]. These testers enabled the construction of P CP s with subconstant soundness, but polynomial size [3, 17, 11]. Previous constructions of almostlinear size testers obtained only constant error [15, 8]. These testers were used to construct almostlinear size LT Cs and almostlinear size P CP s with constant soundness