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 to sample an approximately uniform solution of Φ
(or equivalently, approximate the partition function of Φ).
Let α=m/n be the density. The previous best algorithm runs in time
npoly(k,α) for any α≲2k/300 [Galanis,
Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves
both bounds by providing an almost-linear time sampler for any
α≲2k/3.
The density α captures the \emph{average degree} in the random
formula. In the worst-case model with bounded \emph{maximum degree}, current
best efficient sampler works up to degree bound 2k/5 [He, Wang, and Yin,
FOCS'22 and SODA'23], which is, for the first time, superseded by its
average-case counterpart due to our 2k/3 bound. Our result is the first
progress towards establishing the intuition that the solvability of the
average-case model (random k-CNF formula with bounded average degree) is
better than the worst-case model (standard k-CNF formula with bounded maximal
degree) in terms of sampling solutions.Comment: 51 pages, all proofs added, and bounds slightly improve