Long Range Frustrations in a Spin Glass Model of the Vertex Cover Problem

TL;DR

This paper develops a mean field theory to analyze long range frustrations in a spin glass model of the NP-hard vertex cover problem, providing analytical insights consistent with existing numerical results.

Contribution

It introduces a novel mean field approach to understand long range frustrations in spin glass models applied to the vertex cover problem.

Findings

Analytical ground-state energy density matches numerical results.

Fraction of frozen vertices aligns with previous studies.

Long range frustrations are characterized in the spin glass framework.

Abstract

In a spin glass system on a random graph, some vertices have their spins changing among different configurations of a ground--state domain. Long range frustrations may exist among these unfrozen vertices in the sense that certain combinations of spin values for these vertices may never appear in any configuration of this domain. We present a mean field theory to tackle such long range frustrations and apply it to the NP-hard minimum vertex cover (hard-core gas condensation) problem. Our analytical results on the ground-state energy density and on the fraction of frozen vertices are in good agreement with known numerical and mathematical results.