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The maximum independent union of cliques problem: complexity and exact approaches

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Abstract

Given a simple graph, the maximum independent union of cliques problem is to find a maximum-cardinality subset of vertices such that each connected component of the corresponding induced subgraph is a complete graph. This recently introduced problem allows both cliques and independent sets as feasible solutions and is of significant theoretical and applied interest. This paper establishes the complexity of the problem on several classes of graphs (planar, claw-free, and bipartite graphs), and develops an integer programming formulation and an exact combinatorial branch-and-bound algorithm for solving it. Results of numerical experiments with numerous benchmark instances are also reported.

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Acknowledgements

We would like to thank the two anonymous reviewers for their insightful comments. This material is partially based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. Support by NSF Grant CMMI-1538493 and DOD-ONR Grant N00014-13-1-0635 is also gratefully acknowledged.

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Authors and Affiliations

  1. Department of Statistics and Data Science, The University of Texas at Austin, Austin, TX, 78712, USA

    Zeynep Ertem

  2. Texas A&M University, College Station, TX, USA

    Eugene Lykhovyd, Yiming Wang & Sergiy Butenko

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  1. Zeynep Ertem
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  2. Eugene Lykhovyd
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  4. Sergiy Butenko
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Correspondence to Sergiy Butenko.

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Ertem, Z., Lykhovyd, E., Wang, Y. et al. The maximum independent union of cliques problem: complexity and exact approaches. J Glob Optim 76, 545–562 (2020). https://doi.org/10.1007/s10898-018-0694-2

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