Hamming Approximation of NP Witnesses

Abstract

Given a satisfiable 3-SAT formula, how hard is it to find an assignment to the variables that has Hamming distance at most n=2 to a satisfying assignment? More generally, consider any polynomial-time verifier for any NP-complete language. A d(n)-Hamming- approximation algorithm for the verifier is one that, given any member x of the language, outputs in polynomial time a string a with Hamming distance at most d(n) to some witness w, where (x; w) is accepted by the verifier. Previous results have shown that, if P6NP, every NP-complete language has a verifier for which there is no (n=2 n 2=3+d )-Hamming- approximation algorithm, for various constants d 0. Our main result is that, if P6NP, then every paddable NP-complete language has a verifier that admits no (n=2+ O( p n log n))-Hamming-approximation algorithm. That is, one can't get even half the bits right. We also consider natural verifiers for various well-known NP-complete problems. They do have n=2-Hamming-approximation algorithms, but, if P6NP, have no (n=2 n e )-Hamming-approximation algorithms for any constant e > 0. We show similar results for randomized algorithms.