Summary: Improved low­degree testing and its applications
Sanjeev Arora \Lambda
Princeton University
Madhu Sudan y
IBM Watson Research Center
Abstract
NP = PCP(log n; 1) and related results crucially depend upon the close connection
between the probability with which a function passes a low degree test and the distance
of this function to the nearest degree d polynomial. In this paper we study a test
proposed by Rubinfeld and Sudan [RS93]. The strongest previously known connection
for this test states that a function passes the test with probability ffi for some ffi ? 7=8
iff the function has agreement ß ffi with a polynomial of degree d. We present a new,
and surprisingly strong, analysis which shows that the preceding statement is true for
ffi ø 0:5. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry.
As a consequence we obtain an alternate construction for the following proof sys­
tem: A constant prover 1­round proof system for NP languages in which the verifier
uses O(logn) random bits, receives answers of size O(logn) bits, and has an error prob­
ability of at most 2 \Gamma log 1\Gammaffl n . Such a proof system, which implies the NP­hardness of
approximating Set Cover to
within\Omega\Gammathi n) factors, has already been obtained by Raz

Source: Arora, Sanjeev - Department of Computer Science, Princeton University

Collections: Computer Technologies and Information Sciences