Improved Bounds for Sampling Solutions of Random CNF Formulas

TL;DR

This paper presents an almost-linear time algorithm for sampling solutions of random $k$-CNF formulas at higher densities than previously possible, advancing understanding of average-case versus worst-case complexity.

Contribution

It significantly improves bounds for sampling solutions in random $k$-CNF formulas, achieving near-linear time for densities up to $2^{k/3}$, surpassing prior bounds.

Findings

Achieves almost-linear time sampling for $oldsymbol{ ext{density }oldsymbol{ ho extless 2^{k/3}}}$

Surpasses previous bounds of $oldsymbol{ ext{density }oldsymbol{ ho extless 2^{k/300}}}$

First to show average-case model solvability exceeds worst-case bounds in terms of degree

Abstract

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 to sample an approximately uniform solution of $\Phi$ (or equivalently, approximate the partition function of $\Phi$). Let $\alpha=m/n$ be the density. The previous best algorithm runs in time $n^{\mathsf{poly}(k,\alpha)}$ for any $\alpha\lesssim2^{k/300}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves both bounds by providing an almost-linear time sampler for any $\alpha\lesssim2^{k/3}$. The density $\alpha$ 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sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound.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.