Hamming Approximation of NP Witnesses

TL;DR

This paper proves that, assuming P != NP, it is computationally hard to find approximate solutions close to actual witnesses for NP problems, even within half the bits, highlighting fundamental limits of approximation.

Contribution

It establishes new hardness results for Hamming-approximate NP witnesses, showing no efficient algorithms can achieve close approximations beyond half the bits, including for natural NP-complete problems.

Findings

No polynomial-time algorithm can find a witness within half the bits if P != NP.

Natural NP-complete problem verifiers admit no (n/2 - n^epsilon)-approximation algorithms.

Similar hardness results hold for randomized algorithms.

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 P != NP, then 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 P != NP, then every paddable NP-complete language has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation algorithm. That is, one cannot 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 P != NP, have no (n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0. We show similar results for randomized algorithms.