Improved Bounds for Sampling Solutions of Random CNF Formulas

Abstract

Let Φ be a random k-CNF formula on n variables and m clauses, where each clause is a disjunction of k literals chosen independently and uniformly. Our goal is, for most Φ, to (approximately) uniformly sample from its solution space.Let α = m/n be the density. The previous best algorithm runs in time n poly(k,α) for any α 2 k/300 [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in near-linear time for any α 2 k/3 .