Theory of Computing ------------------- Title : Hamming Approximation of NP Witnesses Authors : Daniel Sheldon and Neal E. Young Volume : 9 Number : 22 Pages : 685-702 URL : https://theoryofcomputing.org/articles/v009a022 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 \ne NP, every NP -complete language has a verifier for which there is no $(n/2-n^{2/3+\delta})$-Hamming-approximation algorithm, for various constants $\delta\ge 0$. Our main result is that, if P \ne NP, then every paddable NP-complete languagehas a verifier that admits no $(n/2+O(\sqrt{n\log n}))$-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 P \ne NP, have no $(n/2-n^\epsilon)$-Hamming-approximation algorithms for any constant $\epsilon>0$. We show similar results for randomized algorithms.