## A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP (1997)

Venue: | IN PROC. 29TH ACM SYMP. ON THEORY OF COMPUTING, 475-484. EL PASO |

Citations: | 286 - 21 self |

### BibTeX

@INPROCEEDINGS{Raz97asub-constant,

author = {Ran Raz and Shmuel Safra},

title = {A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP},

booktitle = {IN PROC. 29TH ACM SYMP. ON THEORY OF COMPUTING, 475-484. EL PASO},

year = {1997},

publisher = {}

}

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### OpenURL

### 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 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...