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 SNP-hard 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 N-vertex graph to within a factor of N ɛ is NP-hard. 1

The study of self-testing and self-correcting programs leads to the search for robust characterizations of functions. Here we make this notion precise and show such a characterization for polynomials. From this characterization, we get the following applications.

We introduce a new low-degree--test, one that uses the restriction of low-degree polynomials to planes (i.e., affine sub-spaces of dimension 2), rather than the restriction to lines (i.e., affine sub-spaces 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 low-error 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 error-probability 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 SET-COVER to within a logarithmic factors is NP-hard. Previous analysis for low-degree-tests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error...

The contribution of this paper is two-fold. First, a connection is shown between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover 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. 1

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 non-approximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight.

We give constructions of probabilistically checkable proofs (PCPs) of length n·poly(log n) (to prove satisfiability of circuits of size n) that can verified by querying poly(log n) bits of the proof. We also give constructions of locally testable codes (LTCs) with similar parameters. Previous constructions of short PCPs (from [5] to [9]) relied extensively on properties of low degree multi-variate polynomials. In contrast, our constructions rely on new problems and techniques revolving around the properties of codes based on high degree polynomials in one variable (also known as Reed-Solomon codes). We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed-Solomon codeword, i.e., a univariate polynomial of specified degree. This reduction is simpler than the corresponding steps in previous reductions, and gives a new alternative to using the popular “sumcheck protocol”. We then give a new PCP for the special task of proving that a function is close to being a Reed-Solomon codeword. This step of the construction is by a self-contained recursion, and the only ingredient needed in the analysis is the bi-variate low-degree test of Polischuk and Spielman [27]. Note that our constructions yield LTCs first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite (and less natural) direction of getting LTCs from PCPs.

A low-degree test is a collection of simple, local rules for checking the proximity of an arbitrary function to a low-degree 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 low-degree polynomial. Low-degree tests play an important role in the development of probabilistically checkable proofs. In this paper we present two improvements to the efficiency of low-degree tests. Our first improvement concerns the smallest field size over which a low-degree 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 low-degree 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.

We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spot-checks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the P-NP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...

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We continue the investigation of locally testable codes, i.e., error-correcting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low-degree multivariate polynomials (which are a special case of tensor codes).