Clustering analysis of the ground-state structure of the vertex-cover problem

TL;DR

This paper investigates the structure of solutions in the vertex cover problem on random graphs, revealing a phase transition at connectivity c=e where the solution landscape shifts from simple to complex, aligning with changes in algorithmic complexity.

Contribution

It introduces a novel numerical analysis of the solution landscape of vertex cover, including an algorithm to compute the backbone without enumerating all solutions, and compares findings with analytical predictions.

Findings

Solution clusters are few and small for c<e.

Number of clusters diverges for c>e, indicating complex landscape.

Results support a phase transition at c=e with replica symmetry breaking.

Abstract

Vertex cover is one of the classical NP-complete problems in theoretical computer science. A vertex cover of a graph is a subset of vertices such that for each edge at least one of the two endpoints is contained in the subset. When studied on Erdos-Renyi random graphs (with connectivity c) one observes a threshold behavior: In the thermodynamic limit the size of the minimal vertex cover is independent of the specific graph. Recent analytical studies show that on the phase boundary, for small connectivities c<e, the system is replica symmetric, while for larger connectivities replica symmetry breaking occurs. This change coincides with a change of the typical running time of algorithms from polynomial to exponential. To understand the reasons for this behavior and to compare with the analytical results, we numerically analyze the structure of the solution landscape. For this purpose, we have also developed an algorithm, which allows the calculation of the backbone, without the need to enumerate all solutions. We study exact solutions found with a Branch-and-Bound algorithm as well as configurations obtained via a Monte Carlo simulation. We analyze the cluster structure of the solution landscape by direct clustering of the states, by analyzing the eigenvalue spectrum of correlation matrices and by using a hierarchical clustering method. All results are compatible with a change at c=e. For small connectivities, the solutions are collected in a finite small number of clusters, while the number of cluster diverges slowly with system size for larger connectivities and replica symmetry breaking, but not 1-RSB, occurs.