The number of satisfying assignments of random 2‐SAT formulas

Abstract

We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to "true" under a uniformly random satisfying assignment. KEYWORDS 2-SAT, Belief Propagation, satisfiability problem 1 INTRODUCTION Background and motivationThe random 2-SAT problem was the first random constraint satisfaction problem whose satisfiability threshold could be pinpointed precisely, an accomplishment attained independently by Chvátal and Reed [15] and Goerdt [31] in 1992. The proofs evince the link between the 2-SAT threshold and the percolation phase transition of a random digraph. This connection subsequently enabled Bollobás, Borgs, Chayes, Kim, and Wilson [12] to identify the size of the scaling window, which matches thatThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Scale-Free Random SAT Instances

“…Giráldez-Cru and Levy [21,22] propose a new model of random formulas that, besides heterogeneity, also consider the notion of locality in formulas. Achlioptas et al [23] computes the number of assignments of uniform random 2-SAT formulas, using the cavity method. Finally, Bläsius et al [24] analyze the hardness of random instances according to their heterogeneity and locality.…”

Counting Solutions to Random CNF Formulas

“…It has been shown that 1 n log Z(\Phi ) is concentrated around its expectation [1,16] for \alpha < (1 -o k (1)) \cdot 2 k ln k/k. However, for the random k-SAT model, there is no known formula for the expectation \BbbE 1 n log Z(\Phi ) (though see [3] and [40,17] for progress along these lines for the case k = 2 and for more symmetric models of random formulas, respectively). Regarding the algorithmic question, Montanari and Shah [36] have given an efficient algorithm to approximate a closely related (permissive) version 1 of 1 n log Z(\Phi ) if \alpha \leq 2 log k k (1 + o k (1)), based on the correlation decay method and the uniqueness threshold of the Gibbs distribution.…”