On integer programming models for the maximum 2-club problem and its robust generalizations in sparse graphs

Abstract

We consider the maximum 2-club problem, which aims at finding an induced subgraph of maximum cardinality with the diameter at most two. Such subgraphs arise from a popular diameter-based clique relaxation concept, as a subgraph is a clique if and only if its diameter is one. In a 2-club every pair of non-adjacent vertices has a common neighbor; this “2-hop” property naturally arises in a variety of applications. In this paper, by exploiting a somewhat different interpretation of the problem, we provide two new mixed-integer programming (MIP) models for finding maximum 2-clubs. Our MIPs provide much tighter linear programming (LP) relaxations for sufficiently sparse graphs and have fewer constraints than the standard integer programming (IP) model at the expense of having slightly more continuous variables. We also consider feasibility versions of our MIPs that verify whether there exists a 2-club of some specified size. Then we incorporate them into a simple-to-implement “feasibility-check” algorithm that iteratively solves one of the feasibility MIPs for each possible 2-club size within some known lower and upper bounds. The upper bound is obtained from an LP relaxation of our new MIPs and is shown to be sharp. Furthermore, we show how to extend our approaches for solving some “robust” (attack- and failure-tolerant) generalizations of the maximum 2-club problem. Finally, we perform an extensive computational study with randomly generated and real-life graphs to support our theoretical results and to provide some empirical observations and insights.