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

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