An alternate view of complexity in k-SAT problems

TL;DR

This paper presents a novel method for determining the satisfiability threshold in k-SAT problems on Bethe lattices, matching predictions from the replica-symmetry breaking approach without relying on its assumptions.

Contribution

It introduces an alternative route to compute the satisfiability threshold that does not depend on the solution-space structure assumptions of the RSB method.

Findings

Exact match with cavity method predictions under 1-RSB

Provides alternative interpretations of RSB equations

No assumptions about solution-space structure needed

Abstract

The satisfiability threshold for constraint satisfaction problems is that value of the ratio of constraints (or clauses) to variables, above which the probability that a random instance of the problem has a solution is zero in the large system limit. Two different approaches to obtaining this threshold have been discussed in the literature - using first or second-moment methods which give rigorous bounds or using the non-rigorous but powerful replica-symmetry breaking (RSB) approach, which gives very accurate predictions on random graphs. In this paper, we lay out a different route to obtaining this threshold on a Bethe lattice. We need make no assumptions about the solution-space structure, a key assumption in the RSB approach. Despite this, our expressions and threshold values exactly match the best predictions of the cavity method under the 1-RSB assumption. Our method hence provides alternate interpretations as well as motivations for the key equations in the RSB approach.