#5020. On integer programming models for the maximum 2-club problem and its robust generalizations in sparse graphs

July 2026publication date
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Journal’s subject area:
Modeling and Simulation;
Management Science and Operations Research;
Industrial and Manufacturing Engineering;
Computer Science (all);
Information Systems and Management;
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More details about the manuscript: Science Citation Index Expanded or/and Social Sciences Citation Index
Abstract:
We consider the maximum 2-club problem, which aims at finding an induced subgraph of maximum cardinality with the diameter at most two. 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. 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. 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.
Keywords:
2-Clubs; Clique relaxations; Graph theory; Integer programming; Networks

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