Upper bounds and heuristics for the 2-club problem

Given an undirected graph G = (V, E), a k-club is a subset of V that induces a subgraph of diameter at most k. The k-club problem is that of finding the maximum cardinality k-club in G. In this paper we present valid inequalities for the 2-club polytope and derive conditions for them to define facets. These inequalities are the basis of a strengthened formulation for the 2-club problem and a cutting plane algorithm. The LP relaxation of the strengthened formulation is used to compute upper bounds on the problem’s optimum and to guide the generation of near-optimal solutions. Numerical experiments indicate that this approach is quite effective in terms of solution quality and speed, especially for low density graphs.