The random K-satisfiability problem: from an analytic solution to an efficient algorithm

TL;DR

This paper analyzes the satisfiability of random K-SAT problems using the cavity method, revealing an intermediate phase that explains algorithm slowdown and leading to the development of a new, efficient solving algorithm.

Contribution

It introduces an analytic phase diagram for 3-SAT using the cavity method and proposes a novel algorithm based on surveys of local fields, improving solution efficiency.

Findings

Identified an intermediate phase with metastable states causing search slowdown

Computed the fundamental order parameter for individual samples

Developed a new algorithm with very good performance on 3-SAT

Abstract

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.