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.