Replica bounds for optimization problems and diluted spin systems

TL;DR

This paper extends a technique to derive bounds from the replica method for diluted spin systems and optimization problems, providing a framework to estimate free energies and thresholds in complex disordered models.

Contribution

It generalizes a recent technique to diluted models and optimization problems, establishing the replica method as a tool for variational bounds in these systems.

Findings

Replica method yields lower bounds for free energy and ground state energy.

Provides upper bounds for satisfiability thresholds in K-SAT.

Applicable to models like Viana-Bray, diluted p-spin, and XOR-SAT.

Abstract

In this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by Guerra (cond-mat/0205123) to prove that the replica method generates variational bounds for disordered systems. We analyze a family of models that includes the Viana-Bray model, the diluted p-spin model or random XOR-SAT problem, and the random K-SAT problem, showing that the replica method provides an improvable scheme to obtain lower bounds of the free-energy at all temperatures and of the ground state energy. In the case of K-SAT the replica method thus gives upper bounds of the satisfiability threshold.