The complexity of unique k-SAT: an isolation lemma for k-CNFs

Abstract

We provide some evidence that unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs and unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k/spl ges/1, s/sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for k-SAT} and, similarly, /spl sigma//sub k/=inf{/spl delta//spl ges/0|/spl exist/aO(2/sup /spl delta/n/)-time randomized algorithm for Unique k-SAT}, we show that lim/sub k/spl rarr//spl infin//s/sub k/=lim/sub k/spl rarr//spl infin///spl sigma//sub k/. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2/sup /spl epsi/n/ for every /spl epsi/>0, then so can k-SAT for k/spl ges/3. Our main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with nontrivial, albeit exponentially small, success probability.