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The Maximum k -Colorable Subgraph Problem and Related Problems

Author

Listed:
  • Olga Kuryatnikova

    (Erasmus University Rotterdam, Rotterdam 3062 PA, Netherlands)

  • Renata Sotirov

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg 5000 LE, Netherlands)

  • Juan C. Vera

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg 5000 LE, Netherlands)

Abstract

The maximum k -colorable subgraph (M k CS) problem is to find an induced k -colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the M k CS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. To simplify these relaxations, we exploit the symmetry arising from permuting the colors, as well as the symmetry of the given graphs when applicable. We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the M k CS problem to derive bounds on the chromatic number of a graph. Our numerical results verify that the proposed relaxations provide strong bounds for the M k CS problem and that those outperform existing bounds for most of the test instances. Summary of Contribution: The maximum k -colorable subgraph (M k CS) problem is to find an induced k -colorable subgraph with maximum cardinality in a given graph. The M k CS problem has a number of applications, such as channel assignment in spectrum sharing networks (e.g., Wi-Fi or cellular), very-large-scale integration design, human genetic research, and so on. The M k CS problem is also related to several other optimization problems, including the graph partition problem and the max- k -cut problem. The two mentioned problems have applications in parallel computing, network partitioning, floor planning, and so on. This paper is an in-depth analysis of the M k CS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. Further, our analysis relates the M k CS results with the stable set and the chromatic number problems. We provide extended numerical results that verify that the proposed bounding approaches provide strong bounds for the M k CS problem and that those outperform existing bounds for most of the test instances. Moreover, our lower bounds on the chromatic number of a graph are competitive with existing bounds in the literature.

Suggested Citation

  • Olga Kuryatnikova & Renata Sotirov & Juan C. Vera, 2022. "The Maximum k -Colorable Subgraph Problem and Related Problems," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 656-669, January.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:1:p:656-669
    DOI: 10.1287/ijoc.2021.1086
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    References listed on IDEAS

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