Approximating satisfiability transition by suppressing fluctuations

TL;DR

This paper introduces a statistical mechanics-inspired method to rigorously estimate upper bounds for the satisfiability transition in random boolean formulas by analyzing core substructures.

Contribution

It presents a novel approach using core identification and self-consistency equations to improve bounds on the satisfiability threshold in complex combinatorial problems.

Findings

Derived self-consistency equations for core parameters

Computed improved annealing bounds for K-XOR-SAT, K-SAT, and positive 1-in-K-SAT

Demonstrated the method's effectiveness on multiple problem types

Abstract

Using methods and ideas from statistical mechanics, we propose a simple method for obtaining rigorous upper bounds for satisfiability transition in random boolean expressions composed of N variables and M clauses with K variables per clause. Determining the location of satisfiability threshold $\alpha_c=M/N$ for a number of difficult combinatorial problems is a major open problem in the theory of random graphs. The method is based on identification of the core -- a subexpression (subgraph) that has the same satisfiability properties as the original expression. We formulate self-consistency equations that determine macroscopic parameters of the core and compute an improved annealing bound. We illustrate the method for three sample problems: K-XOR-SAT, K-SAT and positive 1-in-K-SAT.