Clustering of solutions in the random satisfiability problem

TL;DR

This paper rigorously proves the existence of a clustered phase in the random K-SAT problem for K≥8, confirming predictions made by the cavity method and enhancing understanding of solution space structure.

Contribution

It provides the first rigorous proof of solution clustering in random K-SAT for K≥8, validating key predictions of the cavity method.

Findings

Existence of a clustered phase in random K-SAT for K≥8

Solutions form well-separated clusters in this phase

Results align with cavity method predictions

Abstract

Using elementary rigorous methods we prove the existence of a clustered phase in the random $K$-SAT problem, for $K\geq 8$. In this phase the solutions are grouped into clusters which are far away from each other. The results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks. It can be generalized to other systems of both physical and computational interest.