An improved exponential-time algorithm for k-SAT

Abstract

We propose and analyze a simple new algorithm for finding satisfying assignments of Boolean formulae in conjunctive normal form. The algorithm, ResolveSat, is a randomized variant of the DDL procedure by M. Davis et al. (1962) or Davis-Putnam procedure. Rather than applying the DLL procedure to the input formula F, however; ResolveSat enlarges F by adding additional clauses using limited resolution before performing DLL. The basic idea behind our analysis is the same as by R. Paturi (1997): a critical clause for a variable at a satisfying assignment gives rise to a unit clause in the DLL procedure with sufficiently high probability, thus increasing the probability of finding a satisfying assignment. In the current paper, we analyze the effect of multiple critical clauses (obtained through resolution) in producing unit clauses. We show that, for each k, the running time of ResolveSat on a k-CNF formula is significantly better than 2/sup n/, even in the worst case. In particular we show that the algorithm finds a satisfying assignment of a general 3-CNF in time O(2/sup .446n/) with high probability; where the best previous algorithm has running time O(2/sup .582n/). We obtain a better upper bound of O(2/sup (2ln2-1)/n+0(n))=O(2/sup 0.387n/) for 3-CNF that have at most one satisfying assignment (unique k-SAT). For each k, the bounds for general k-CNF are the best known for the worst-case complexity of finding a satisfying solution for k-SAT, the idea of succinctly encoding satisfying solutions can be applied to obtain lower bounds on circuit site. Here, we exhibit a function f such that any depth-3 AND-OR circuit with bottom fan-in bounded by k requires /spl Omega/(2(c/sub k/n/k)) gates (with c/sub k/>1). This is the first such lower bound with c/sub k/>1.