Vertex cover problem studied by cavity method: Analytics and population dynamics

TL;DR

This paper applies the cavity method to analyze the vertex cover problem on random graphs, deriving analytical and numerical insights into the ground state energy, backbone size, and phase transition behavior.

Contribution

It introduces a cavity method approach to study the vertex cover problem, providing analytical estimates and population dynamics simulations for finite connectivity graphs.

Findings

Ground state energy is finite for c>e as y approaches infinity.

Population dynamics estimates match known bounds and numerical results.

Backbone size is explicitly calculated.

Abstract

We study the vertex cover problem on finite connectivity random graphs by zero-temperature cavity method. The minimum vertex cover corresponds to the ground state(s) of a proposed Ising spin model. When the connectivity c>e=2.718282, there is no state for this system as the reweighting parameter y, which takes a similar role as the inverse temperature \beta in conventional statistical physics, approaches infinity; consequently the ground state energy is obtained at a finite value of y when the free energy function attains its maximum value. The minimum vertex cover size at given c is estimated using population dynamics and compared with known rigorous bounds and numerical results. The backbone size is also calculated.