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

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 | ∃ a O(2)-time randomized algorithm for k-SAT} and, similarly, σk = inf{δ > 0 | ∃ a O(2)-time randomized algorithm for Unique k-SAT}, we show that limk→∞ sk = limk→∞ σk. As a corollary, we prove that, if Unique 3-SAT can be solved in time 2 n for every > 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 kCNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with non-trivial, albeit exponentially small, success probability.