Polyhedral Properties of the Induced Cluster Subgraphs

Abstract

A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph G, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope associated with G is defined as the convex hull of the incidence vectors of all IUCs in the graph. The Maximum IUC problem, which is to find a maximum-cardinality IUC in a graph, finds applications in network-based data analysis. In this paper, we derive several families of facet-defining valid inequalities for the IUC polytope. We also give a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problem for most of the considered families of valid inequalities and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the Maximum IUC problem through computational experiments.

A tight approximation algorithm for the cluster vertex deletion problem

“…For polyhedral results, [14] gives some facet-defining inequalities of a polytope that is affinely equivalent to the Cluster-VD polytope, as well as complete linear descriptions for special classes of graphs.…”

A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

“…For polyhedral results, [15] gives some facet-defining inequalities of the Cluster-VD polytope, as well as complete linear descriptions for special classes of graphs.…”