An improved exponential-time algorithm for k-SAT

We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. Currently, this is the fastest known probabilistic algorithm for k-CNF satisfiability for k ≥ 4 (with a running time of O(2^0.5625n) for 4-CNF). In addition, it is the fastest known probabilistic algorithm for k-CNF, k ≥ 3, that have at most one satisfying assignment (unique k-SAT) (with a running time O(2^{(2 ln 2 − 1)n + o(n)}) = O(2^{0.386 … n}) in the case of 3-CNF). The analysis of the algorithm also gives an upper bound on the number of the codewords of a code defined by a k-CNF. This is applied to prove a lower bounds on depth 3 circuits accepting codes with nonconstant distance. In particular we prove a lower bound Ω(2^{1.282…√>i<n>/i<}) for an explicitly given Boolean function of n variables. This is the first such lower bound that is asymptotically bigger than 2^{√>i<n>/i< + o(√>i<n>/i<)}.