Counting solutions to random CNF formulas

TL;DR

This paper introduces an efficient algorithm for approximately counting solutions in random k-SAT formulas at high densities, surpassing previous methods limited by correlation decay techniques.

Contribution

It presents the first algorithm capable of handling high-density random k-SAT formulas by leveraging a recent technique to manage complex variable correlations.

Findings

Algorithm works efficiently at exponential densities in k

Outperforms previous correlation decay-based methods

Successfully manages high-degree variable correlations

Abstract

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 for the permissive version of the model 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.