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A Branch and Cut solver for the maximum stable set problem

Author

Listed:
  • Steffen Rebennack

    (University of Florida)

  • Marcus Oswald

    (Ruprecht-Karls Universität Heidelberg)

  • Dirk Oliver Theis

    (OvG-Universität Magdeburg)

  • Hanna Seitz

    (Ruprecht-Karls Universität Heidelberg)

  • Gerhard Reinelt

    (Ruprecht-Karls Universität Heidelberg)

  • Panos M. Pardalos

    (University of Florida)

Abstract

This paper deals with the cutting-plane approach to the maximum stable set problem. We provide theoretical results regarding the facet-defining property of inequalities obtained by a known project-and-lift-style separation method called edge-projection, and its variants. An implementation of a Branch and Cut algorithm is described, which uses edge-projection and two other separation tools which have been discussed for other problems: local cuts (pioneered by Applegate, Bixby, Chvátal and Cook) and mod-k cuts. We compare the performance of this approach to another one by Rossi and Smiriglio (Oper. Res. Lett. 28:63–74, 2001) and discuss the value of the tools we have tested.

Suggested Citation

  • Steffen Rebennack & Marcus Oswald & Dirk Oliver Theis & Hanna Seitz & Gerhard Reinelt & Panos M. Pardalos, 2011. "A Branch and Cut solver for the maximum stable set problem," Journal of Combinatorial Optimization, Springer, vol. 21(4), pages 434-457, May.
    • Handle: RePEc:spr:jcomop:v:21:y:2011:i:4:d:10.1007_s10878-009-9264-3
    DOI: 10.1007/s10878-009-9264-3
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    References listed on IDEAS

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    1. Alessandro Avenali, 2007. "Resolution Branch and Bound and an Application: The Maximum Weighted Stable Set Problem," Operations Research, INFORMS, vol. 55(5), pages 932-948, October.
    2. Sven de Vries & Rakesh V. Vohra, 2003. "Combinatorial Auctions: A Survey," INFORMS Journal on Computing, INFORMS, vol. 15(3), pages 284-309, August.
    3. Dawn M. Strickland & Earl Barnes & Joel S. Sokol, 2005. "Optimal Protein Structure Alignment Using Maximum Cliques," Operations Research, INFORMS, vol. 53(3), pages 389-402, June.
    4. E. C. Sewell, 1998. "A Branch and Bound Algorithm for the Stability Number of a Sparse Graph," INFORMS Journal on Computing, INFORMS, vol. 10(4), pages 438-447, November.
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    Cited by:

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    3. Chen, Liang & Chen, Sheng-Jie & Chen, Wei-Kun & Dai, Yu-Hong & Quan, Tao & Chen, Juan, 2023. "Efficient presolving methods for solving maximal covering and partial set covering location problems," European Journal of Operational Research, Elsevier, vol. 311(1), pages 73-87.
    4. Yang Wang & Jin-Kao Hao & Fred Glover & Zhipeng Lü & Qinghua Wu, 2016. "Solving the maximum vertex weight clique problem via binary quadratic programming," Journal of Combinatorial Optimization, Springer, vol. 32(2), pages 531-549, August.
    5. David Bergman & Andre A. Cire & Willem-Jan van Hoeve & J. N. Hooker, 2014. "Optimization Bounds from Binary Decision Diagrams," INFORMS Journal on Computing, INFORMS, vol. 26(2), pages 253-268, May.
    6. Coniglio, Stefano & Furini, Fabio & San Segundo, Pablo, 2021. "A new combinatorial branch-and-bound algorithm for the Knapsack Problem with Conflicts," European Journal of Operational Research, Elsevier, vol. 289(2), pages 435-455.
    7. Stefano Coniglio & Stefano Gualandi, 2022. "Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1006-1023, March.
    8. Ferrarini, Luca & Gualandi, Stefano, 2022. "Total Coloring and Total Matching: Polyhedra and Facets," European Journal of Operational Research, Elsevier, vol. 303(1), pages 129-142.
    9. Wu, Qinghua & Hao, Jin-Kao, 2015. "A review on algorithms for maximum clique problems," European Journal of Operational Research, Elsevier, vol. 242(3), pages 693-709.
    10. Balasundaram, Balabhaskar & Borrero, Juan S. & Pan, Hao, 2022. "Graph signatures: Identification and optimization," European Journal of Operational Research, Elsevier, vol. 296(3), pages 764-775.
    11. Nathan Sudermann‐Merx & Steffen Rebennack & Christian Timpe, 2021. "Crossing Minimal Edge‐Constrained Layout Planning using Benders Decomposition," Production and Operations Management, Production and Operations Management Society, vol. 30(10), pages 3429-3447, October.
    12. Figueiredo, Rosa & Frota, Yuri, 2014. "The maximum balanced subgraph of a signed graph: Applications and solution approaches," European Journal of Operational Research, Elsevier, vol. 236(2), pages 473-487.
    13. Nasirian, Farzaneh & Mahdavi Pajouh, Foad & Balasundaram, Balabhaskar, 2020. "Detecting a most closeness-central clique in complex networks," European Journal of Operational Research, Elsevier, vol. 283(2), pages 461-475.

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