Statistical mechanics of the vertex-cover problem

TL;DR

This paper reviews the application of statistical mechanics methods to analyze the vertex-cover problem, revealing phase transitions and solution complexity on random graphs, and extending understanding to various graph ensembles.

Contribution

It introduces a statistical mechanics framework for vertex-cover, providing exact phase diagrams for certain connectivities and analyzing algorithmic complexity.

Findings

Phase transition in coverability on random graphs.

Exact phase diagram for connectivities c<e.

Full replica symmetry breaking for c>e.

Abstract

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur. We explain a statistical mechanics approach, which works by mapping VC to a hard-core lattice gas, and then applying techniques like the replica trick or the cavity approach. Using these methods, the phase diagram of VC could be obtained exactly for connectivities $c<e$, where VC is replica symmetric. Recently, this result could be confirmed using traditional mathematical techniques. For $c>e$, the solution of VC exhibits full replica symmetry breaking. The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for VC. Finally, we describe recent results for VC when studied on other ensembles of finite- and infinite-dimensional graphs.