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

摘要

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, sk=inf{δ⩾0|∃aO(2δn)-time randomized algorithm fork-SAT} and, similarly, σk=inf{δ⩾0|∃aO(2δn)-time randomized algorithm for Uniquek-SAT}, we show that limk→∞sk=limk→∞σk. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2ϵn for every ϵ>0, 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.