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Algorithms for the maximum k-club problem in graphs

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Abstract

Given a simple undirected graph G, a k-club is a subset of vertices inducing a subgraph of diameter at most k. The maximum k-club problem (MkCP) is to find a k-club of maximum cardinality in G. These structures, originally introduced to model cohesive subgroups in social network analysis, are of interest in network-based data mining and clustering applications. The maximum k-club problem is NP-hard, moreover, determining whether a given k-club is maximal (by inclusion) is NP-hard as well. This paper first provides a sufficient condition for testing maximality of a given k-club. Then it proceeds to develop a variable neighborhood search (VNS) heuristic and an exact algorithm for MkCP that uses the VNS solution as a lower bound. Computational experiments with test instances available in the literature show that the proposed algorithms are very effective on sparse instances and outperform the existing methods on most dense graphs from the testbed.

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Acknowledgements

The authors are thankful to B. Balasundaram and F. Mahdavi for providing the B&B code. Suggestions of anonymous referees that helped to improved the paper are also gratefully acknowledged. This research was partially supported by Air Force Office of Scientific Research (Grant FA9550-09-1-0154) and the US Department of Energy (Grant DE-SC0002051). This research was supported by AFOSR Award No. FA9550-08-1-0483.

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  1. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA

    Shahram Shahinpour & Sergiy Butenko

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  1. Shahram Shahinpour
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  2. Sergiy Butenko
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Correspondence to Shahram Shahinpour.

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Shahinpour, S., Butenko, S. Algorithms for the maximum k-club problem in graphs. J Comb Optim 26, 520–554 (2013). https://doi.org/10.1007/s10878-012-9473-z

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