The complexity of Unique k-SAT: An Isolation Lemma for k-CNFs

  • Authors:

  • Chris Calabro;Russell Impagliazzo;Valentine Kabanets;Ramamohan Paturi

  • Affiliations:

  • University of California, San Diego, USA;University of California, San Diego, USA;Simon Fraser University, Canada;University of California, San Diego, USA

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2008

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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 with at most k literals in each clause and Unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k=1, s"k=inf{@d=0|@?aO(2^@d^n)-time randomized algorithm fork-SAT} and, similarly, @s"k=inf{@d=0|@?aO(2^@d^n)-time randomized algorithm for Uniquek-SAT}, we show that lim"k"-"~s"k=lim"k"-"~@s"k. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2^@e^n for every @e0, then so can k-SAT for all k=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 non-trivial, albeit exponentially small, success probability.