68Q15-NPcomplete.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{NPcomplete}
\pmcreated{2013-03-22 13:01:49}
\pmmodified{2013-03-22 13:01:49}
\pmowner{Henry}{455}
\pmmodifier{Henry}{455}
\pmtitle{NP-complete}
\pmrecord{4}{33429}
\pmprivacy{1}
\pmauthor{Henry}{455}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{68Q15}
\pmsynonym{NP complete}{NPcomplete}
\pmsynonym{NP hard}{NPcomplete}
\pmdefines{NP-hard}

\endmetadata

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\begin{document}
A problem $\pi\in\mathcal{NP}$ is $\mathcal{NP}$ \emph{\PMlinkescapetext{complete}} if for any $\pi^\prime\in\mathcal{NP}$ there is a Cook reduction of $\pi^\prime$ to $\pi$.  Hence if $\pi\in\mathcal{P}$ then every $\mathcal{NP}$ problem would be in $\mathcal{P}$.  A slightly stronger definition requires a Karp reduction or Karp reduction of corresponding decision problems as appropriate.

A search problem $R$ is \emph{$\mathcal{NP}$ hard} if for any $R^\prime\in\mathcal{NP}$ there is a Levin reduction of $R^\prime$ to $R$.
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