Counting Solutions to Random CNF Formulas

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

Certifying solution geometry in random CSPs: counts, clusters and balance

An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: what does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1. Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random $d$-regular graph. 2. Clusters. For Boolean $3$CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3. Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of $+1$s deviates significantly from $50\%$. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal

Belief propagation on the random k-SAT model

Corroborating a prediction from statistical physics, we prove that the belief propagation message passing algorithm approximates the partition function of the random k-SAT model well for all clause/variable densities and all inverse temperatures for which a modest absence of long-range correlations condition is satisfied. This condition is known as “replica symmetry” in physics language. From this result we deduce that a replica symmetry breaking phase transition occurs in the random k-SAT model at low temperature for clause/variable densities below but close to the satisfiability threshold.</p