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 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 { δ ⩾ 0 | ∃ a O ( 2 δ n ) -time randomized algorithm for k -SAT } and, similarly, σ k = inf { δ ⩾ 0 | ∃ a O ( 2 δ n ) -time randomized algorithm for Unique k -SAT } , we show that lim k → ∞ s k = lim k → ∞ σ k . As a corollary, we prove that, if Unique 3-SAT can be solved in time 2 ϵ n 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.