Results 1  10
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3,176
Two Query PCP with SubConstant Error
, 2008
"... We show that the N PComplete language 3SAT has a PCP verifier that makes two queries to a proof of almostlinear size and achieves subconstant probability of error o(1). The verifier performs only projection tests, meaning that the answer to the first query determines at most one accepting answer ..."
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Cited by 57 (6 self)
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to the second query. Previously, by the parallel repetition theorem, there were PCP Theorems with twoquery projection tests, but only (arbitrarily small) constant error and polynomial size [29]. There were also PCP Theorems with subconstant error and almostlinear size, but a constant number of queries
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 339 (20 self)
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, 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
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 778 (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 .
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 498 (68 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query
Subconstant error probabilistically checkable proof of almost linear size
 Electronic Colloquium on Computational Complexity (ECCC
"... We show a construction of a PCP with both subconstant error and almostlinear size. Specifically, for some constant 0 < α < 1, we construct a PCP verifier for checking satisfiability of Boolean formulas that on input of size n uses logn+O((log n)1−α) random bits to query a constant number of ..."
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Cited by 10 (4 self)
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We show a construction of a PCP with both subconstant error and almostlinear size. Specifically, for some constant 0 < α < 1, we construct a PCP verifier for checking satisfiability of Boolean formulas that on input of size n uses logn+O((log n)1−α) random bits to query a constant number
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 ..."
Abstract

Cited by 8 (0 self)
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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
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 169 (9 self)
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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
A SubConstant ErrorProbability PCP Characterization of NP  PART II: The Consistency Test
 29th ACM Symposium on the Theory of Computing
, 1996
"... This paper introduces a new consistencytest for a class of codes, referred to as geometriccodes, and proves the test to be of small errorprobability. This consistencytest enables us to conclude a strong characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ..."
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Cited by 7 (3 self)
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of PCP have managed to achieve, with constant number of accesses, errorprobability of, at best, a constant. Lowdegree polynomials are a special case of geometriccodes, hence our consistencytest implies a lowdegreetest, which is the first to exhibit subconstant probability of error. The proof
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 17 (5 self)
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.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
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