We introduce a new low-degree-test, one that uses the restriction of low-degree polynomials to planes (i. e., afine sub-spaces of dimension 2), rather than the restriction to lines (i. e., afine sub-spaces of dimension 1). We prove the new test to be of a very small emorprobability (in particular, much smaller than constant). The new test enables us to prove a low-error characterization of NP in terms of PCP. Specifically, OUT theorem states that, for any given c > 0, membership in any NP language can be verijied with 0(1) accesses, each r'eading logarithmic number of bits, and such that the error-probability is 2-'"~'-' 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 characten"zations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error-probability of, at best, a constant. The proof for the smail err-or-probability of our new low-degree-test is, nevertheless, significantly simpler than previous proofs. In particular, it is combinatorial and geometrical in nature, rather than algebraic. R. Rubinfeld. A mathematical theory o,jselj-checkingj self-testing and self-correctingPrograms. PhD thesis, U.C. Berkeley, 1990. A. Shamir. fp = PSPACE.

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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 NP-hard.

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We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.

This work addresses k-restriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Σ m that satisfies a given set of k-wise demands. For every k positions and every demand, there must be at least one string in the list that satisfies the demand at these positions. Problems of this form frequently arise in different fields in Computer Science.The standard approach for deterministically solving such problems is via almost k-wise independence or k-wise approximations for other distributions. We offer a generic algorithmic method that yields considerably smaller constructions. To this end, we generalize a previous work of Naor, Schulman and Srinivasan [18]. Among other results, we greatly enhance the combinatorial objects in the heart of their method, called splitters, and construct multi-way splitters, using a new discrete version of the topological Necklace Splitting Theorem [1].We utilize our methods to show improved constructions for group testing [19] and generalized hashing [3], and an improved inapproximability result for Set-Cover under the assumption P = N P.

The contribution of this paper is twofold. First, a connection is established 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 eftlcient multi-prover 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.

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