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.
We describe recent work on a variant of a distance labeling problem in graphs, called the forbidden-set labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.