An improved exponential-time algorithm for k-SAT

• Authors: Ramamohan Paturi;Pavel Pudlák;Michael E. Saks;Francis Zane
• Venue: Journal of the ACM (JACM)
• Year: 2005

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(20.5625n) 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(20.386 … 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(21.282…√i/i) for an explicitly given Boolean function of n variables. This is the first such lower bound that is asymptotically bigger than 2√i/io(√i/i.