Theory of Computing
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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
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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.