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Continuous quadratic programming formulations of optimization problems on graphs

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  • Hager, William W.
  • Hungerford, James T.

Abstract

Four NP-hard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic programming formulations for the edge separator problem, the maximum clique problem, and the maximum independent set problem. All of these formulations, when expressed as maximization problems, are shown to follow from the convexity properties of the objective function along the edges of the feasible set. An algorithm is given which exploits the continuous formulation of the vertex separator problem to quickly compute approximate separators. Computational results are given.

Suggested Citation

  • Hager, William W. & Hungerford, James T., 2015. "Continuous quadratic programming formulations of optimization problems on graphs," European Journal of Operational Research, Elsevier, vol. 240(2), pages 328-337.
    • Handle: RePEc:eee:ejores:v:240:y:2015:i:2:p:328-337
    DOI: 10.1016/j.ejor.2014.05.042
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    References listed on IDEAS

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    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
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    3. Luana E. Gibbons & Donald W. Hearn & Panos M. Pardalos & Motakuri V. Ramana, 1997. "Continuous Characterizations of the Maximum Clique Problem," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 754-768, August.
    4. S. Lucidi & F. Rinaldi, 2010. "Exact Penalty Functions for Nonlinear Integer Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 145(3), pages 479-488, June.
    5. Mohamed Didi Biha & Marie-Jean Meurs, 2011. "An exact algorithm for solving the vertex separator problem," Journal of Global Optimization, Springer, vol. 49(3), pages 425-434, March.
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    Cited by:

    1. William W. Hager & James T. Hungerford & Ilya Safro, 2018. "A multilevel bilinear programming algorithm for the vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 189-223, January.
    2. Benlic, Una & Epitropakis, Michael G. & Burke, Edmund K., 2017. "A hybrid breakout local search and reinforcement learning approach to the vertex separator problem," European Journal of Operational Research, Elsevier, vol. 261(3), pages 803-818.

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