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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 718 (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. 1
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 365 (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.
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 145 (18 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.
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 128 (9 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 ∈
Linearity testing in characteristic two
 IEEE Transactions on Information Theory
, 1996
"... The case we are interested in is when the underlying groups are G=GF(2)n and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(f) measures its distance to a Hadamard code (nor ..."
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Cited by 57 (7 self)
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The case we are interested in is when the underlying groups are G=GF(2)n and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(f) measures its distance to a Hadamard code (normalized so as to be a real number between 0 and 1). The quantity Err(f) is a parameter that is "easy to measure " and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation results. In this context, improved analyses translate into better nonapproximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight.
Assignment testers: Towards a combinatorial proof of the PCP theorem
 SIAM Journal on Computing
, 2004
"... In this work we look back into the proof of the PCP Theorem, with the goal of finding new proofs that are “more combinatorial ” and arguably simpler. For that we introduce the notion of an assignment tester, which is a strengthening of the standard PCP verifier, in the following sense. Given a state ..."
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Cited by 24 (3 self)
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In this work we look back into the proof of the PCP Theorem, with the goal of finding new proofs that are “more combinatorial ” and arguably simpler. For that we introduce the notion of an assignment tester, which is a strengthening of the standard PCP verifier, in the following sense. Given a statement and an alleged proof for it, while the PCP verifier checks correctness of the statement, the assignmenttester checks correctness of the statement and the proof. This notion enables composition that is truly modular, i.e., one can compose two assignmenttesters without any assumptions on how they are constructed. A related notion called PCPs of Proximity was independently introduced in [BenSasson et. al. STOC 04]. We provide a toolkit of (nontrivial) generic transformations on assignment testers. These transformations may be interesting in their own right, and allow us to present the following two main results: 1. The first is a new proof of the PCP Theorem. This proof relies on a rather weak assignment tester given as a “black box”. From this, we construct combinatorially the full PCP. An important component of this proof is a new combinatorial aggregation technique (i.e., a new transformation that allows the verifier to read fewer, though possibly longer, “pieces ” of the proof). An implementation of the blackbox tester can be obtained from the algebraic proof techniques that already appear in [BFLS91, FGL + 91]. Obtaining a combinatorial implementation of this tester would give a purely combinatorial proof for the PCP theorem, which we view as an interesting open problem. 2. Our second construction is a “standalone ” combinatorial construction showing NP ⊆ P CP [polylog, 1]. This implies, for example, that approximating maxSAT is quasiNPhard. This construction relies on a transformation that makes an assignment tester “oblivious”: so that the proof locations read are independent of the statement that is being proven. This eliminates, in a rather surprising manner, the need for aggregation in a crucial point in the proof. 1
On the Robustness of Functional Equations
 SIAM Journal on Computing
, 1994
"... In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for ..."
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Cited by 22 (2 self)
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In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which are of the form 8x; y F [f(x \Gamma y); f(x + y); f(x); f(y)] = 0, where F is an algebraic function. We give conditions on such functional equations that imply robustness. Our results have applications to the area of selftesting/correcting programs. We show that selftesters and selfcorrectors can be found for many functions satisfying robust functional equations, including algebraic functions of trigonometric functions such as tan x; 1 1+cotx ; Ax 1\GammaAx ; cosh x. 1 Introduction The mathematical field of functional equations is concerned with the following prototypical problem: Given a set of properties (fun...
Probabilistically Checkable Proofs and Codes
"... Abstract. NP is the complexity class of problems for which it is easy to check that a solution is correct. In contrast, finding solutions to certain NP problems is widely believed to be hard. The canonical example is the sat problem: given a Boolean formula, it is notoriously difficult to come up wi ..."
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Abstract. NP is the complexity class of problems for which it is easy to check that a solution is correct. In contrast, finding solutions to certain NP problems is widely believed to be hard. The canonical example is the sat problem: given a Boolean formula, it is notoriously difficult to come up with a satisfying assignment, whereas given a proposed assignment it is trivial to plug in the values and verify its correctness. Such an assignment is an “NPproof ” for the satisfiability of the formula. Although the verification is simple, it is not local, i.e., a verifier must typically read (almost) the entire proof in order to reach the right decision. In contrast, the landmark PCP theorem [4, 3] says that proofs can be encoded into a special “PCP ” format, that allows speedy verification. In the new format it is guaranteed that a PCP proof of a false statement will have many many errors. Thus such proofs can be verified by a randomized procedure that is local: it reads only a constant (!) number of bits from the proof and with high probability detects an error if one exists. How are these PCP encodings constructed? First, we describe the related and possibly cleaner problem of constructing locally testable codes. These are essentially error correcting codes that are testable by a randomized local algorithm. We point out some connections between local testing and questions about stability of various mathematical systems. We then sketch two known ways of constructing PCPs.
Public Key Locally Decodable Codes with Short Keys ∗
, 2011
"... This work considers locally decodable codes in the computationally bounded channel model. The computationally bounded channel model, introduced by Lipton in 1994, views the channel as an adversary which is restricted to polynomialtime computation. Assuming the existence of INDCPA secure publickey ..."
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This work considers locally decodable codes in the computationally bounded channel model. The computationally bounded channel model, introduced by Lipton in 1994, views the channel as an adversary which is restricted to polynomialtime computation. Assuming the existence of INDCPA secure publickey encryption, we present a construction of publickey locally decodable codes, with constant codeword expansion, tolerating constant error rate, with locality O(λ), and negligible probability of decoding failure, for security parameter λ. Hemenway and Ostrovsky gave a construction of locally decodable codes in the publickey model with constant codeword expansion and locality O(λ2), but their construction had two major drawbacks. The keys in their scheme were proportional to n, the length of the message, and their schemes were based on the Φhiding assumption. Our keys are of length proportional to the security parameter instead of the message, and our construction relies only on the existence of INDCPA secure encryption rather than on specific numbertheoretic assumptions. Our scheme also decreases the locality from O(λ2) to O(λ). Our construction can be