We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

中文翻译：

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随机 CNF 公式的计数解

我们给出了第一个有效算法，当公式的密度随 $k$ 呈指数缩放时，可以近似计算随机 $k$-SAT 模型中的解数。以前最好的计数算法归功于 Montanari 和 Shah，并且基于相关衰减方法，该方法适用于密度 $(1+o_k(1))\frac{2\log k}{k}$，Gibbs 唯一性模型的阈值。相反，我们的算法利用 Moitra 的最新技术来处理随机公式。我们设置中的主要挑战是考虑到高阶变量的存在，这些变量的边际分布难以控制并且会导致公式内的显着相关性。