On the Complexity of Finding Narrow Proofs

We study the complexity of the following "resolution width problem": Does a given 3-CNF formula have a resolution refutation of width k? For fixed k, refutations of width k can easily be found in polynomial time. We prove a matching polynomial lower bound for the resolution width problem that shows that there is no significant faster way to decide the existence of a width-k refutation than exhaustively searching for it. This lower bound is unconditional and does not rely on any unproven complexity theoretic assumptions. We also prove that the resolution width problem is EXPTIME-complete (if k is part of the input). This confirms a conjecture by Vardi, who has first raised the question for the complexity of the resolution width problem. Furthermore, we prove that the variant of the resolution width problem for regular resolution is PSPACE-complete, confirming a conjecture by Urquhart.