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sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound.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.

翻译：设$\Phi$为包含$n$个变量和$m$个子句的随机$k$-CNF公式，其中每个子句由独立均匀选取的$k$个文字的析取构成。我们的目标是采样$\Phi$的近似均匀解（或等价地，近似$\Phi$的配分函数）。令$\alpha=m/n$表示密度。此前最优算法在$\alpha\lesssim2^{k/300}$时的时间复杂度为$n^{\mathsf{poly}(k,\alpha)}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]。我们的结果将采样器的适用范围显著提升至$\alpha\lesssim2^{k/3}$，同时实现近乎线性的运行时间。密度$\alpha$刻画了随机公式的\textit{平均度数}。在最大度数有界的最坏情形模型中，当前最高效采样器仅能处理度数不超过$2^{k/5}$的情形 [He, Wang, and Yin, FOCS'22 and SODA'23]，而我们的$2^{k/3}$界首次使得平均情形模型超越了最坏情形模型。该结果标志着向以下直觉认知迈出第一步：在解采样问题上，平均情形模型（平均度数有界的随机$k$-CNF公式）的可解性优于最坏情形模型（最大度数有界的标准$k$-CNF公式）。