neal young / Sheldon13Hamming

  • Theory of Computing 9(22):685-702(2013)
    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, then 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 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$\ne$NP, have no $(n/2-n^\epsilon)$-Hamming-approximation algorithms for any constant $\epsilon>0$.

    We show similar results for randomized algorithms.
    First draft circulated in 2003.

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