Improved Bounds for Sampling Solutions of Random CNF Formulas

Let Phi 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 Phi, to (approximately) uniformly sample from its solution space. Let alpha = m/n be the density. The previous best algorithm runs in time n(poly(k,alpha)) for any alpha less than or similar to 2(k/300) [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in almost-linear time for any alpha less than or similar to 2(k/3).