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 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|∃aO(2^δn)-time randomized algorithm for k-SAT} and, similarly, σ_k=inf{δ≥0|∃aO(2^δn)-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^εn for every ε>0, then so can k-SAT for 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 nontrivial, albeit exponentially small, success probability.