Counting good truth assignments of random k-SAT formulae

TL;DR

This paper introduces a polynomial-time deterministic algorithm to approximate the number of nearly satisfying truth assignments in random k-SAT formulas, leveraging belief propagation and interpolation methods.

Contribution

It presents a novel approximation algorithm for counting good truth assignments in random k-SAT, surpassing traditional MCMC-based methods and establishing a new threshold for Gibbs distribution uniqueness.

Findings

Algorithm achieves small relative error with high probability

Threshold for Gibbs distribution uniqueness is derived as 2k^{-1} log k

Method extends to large random k-SAT instances

Abstract

We present a deterministic approximation algorithm to compute logarithm of the number of `good' truth assignments for a random k-satisfiability (k-SAT) formula in polynomial time (by `good' we mean that violate a small fraction of clauses). The relative error is bounded above by an arbitrarily small constant epsilon with high probability as long as the clause density (ratio of clauses to variables) alpha<alpha_{u}(k) = 2k^{-1}\log k(1+o(1)). The algorithm is based on computation of marginal distribution via belief propagation and use of an interpolation procedure. This scheme substitutes the traditional one based on approximation of marginal probabilities via MCMC, in conjunction with self-reduction, which is not easy to extend to the present problem. We derive 2k^{-1}\log k (1+o(1)) as threshold for uniqueness of the Gibbs distribution on satisfying assignment of random infinite tree k-SAT formulae to establish our results, which is of interest in its own right.