# DIMACS Princeton Theory Seminar

## Title:

Sub-Constant Error PCP Characterization of NP

## Speaker:

Ran Raz
Weizmann Institute

## Place:

Room 402, Computer Science Building
35 Olden Street
Princeton University.

## Time:

12:05 PM (Lunch will be served at 11:45 AM)
Thursday, April 24, 1997 - Note change of day

## Contact:

Sanjeev Arora (arora@cs.princeton.edu)

## Abstract:

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 error-probability (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 $\epsilon > 0$, 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 $2^{-\log^{1 -\epsilon}n}$. Our results are in fact stronger.

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-probability of, at best, a constant. The proof for the small error-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.

Joint work with S. Safra (Tel-Aviv University)