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Given a satisfiable 3-SAT formula, how hard is it to find an assignment tothe variables that has Hamming distance at most n/2 to a satisfying assignment?More generally, consider any polynomial-time verifier for any NP-completelanguage. 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 withHamming distance at most d(n) to some witness w, where (x,w) is accepted by theverifier. Previous results have shown that, if P != NP, then every NP-completelanguage 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 languagehas a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximationalgorithm. That is, one cannot get even half the bits right. We also considernatural verifiers for various well-known NP-complete problems. They do haven/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.

Hamming Approximation of NP Witnesses