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Preprocessing and valid inequalities for exact detection of critical nodes via integer programming

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  • Sheng-Jie Chen

    (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Liang Chen

    (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Guang-Ming Li

    (Beijing University of Posts and Telecommunications)

  • Yu-Hong Dai

    (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

Abstract

The critical nodes detection problem (CNDP) involves identifying a limited number of nodes for removal from an undirected graph, to maximize the disconnections between remaining node pairs. In this paper, we shall provide a high-efficiency algorithm for precisely solving the integer programming (IP) formulations for the CNDP. Firstly, a preprocessing procedure is introduced, which can not only reduce the size of the exponential-size IP formulation of the problem but also strengthen the linear programming relaxation. Secondly, the polyhedral properties of the polytope associated with the exponential-size IP formulation are explored, providing a flexible way to derive facet-defining inequalities for the polytope from certain projected ones. Thirdly, a family of strong valid inequalities based on clique subgraphs is developed for the polytope, with both necessary and sufficient conditions for them to be facet-defining. The complexity and algorithm of the separation problem for these inequalities are also investigated. Finally, we extend our research findings from the exponential-size IP formulation to two polynomial-size IP reformulations for the CNDP. Computational results demonstrate the efficacy of incorporating our proposed preprocessing and valid inequalities into an IP solver for solving all three CNDP formulations.

Suggested Citation

  • Sheng-Jie Chen & Liang Chen & Guang-Ming Li & Yu-Hong Dai, 2025. "Preprocessing and valid inequalities for exact detection of critical nodes via integer programming," Computational Optimization and Applications, Springer, vol. 92(1), pages 215-263, September.
    • Handle: RePEc:spr:coopap:v:92:y:2025:i:1:d:10.1007_s10589-025-00698-5
    DOI: 10.1007/s10589-025-00698-5
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    References listed on IDEAS

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    1. Bernardetta Addis & Roberto Aringhieri & Andrea Grosso & Pierre Hosteins, 2016. "Hybrid constructive heuristics for the critical node problem," Annals of Operations Research, Springer, vol. 238(1), pages 637-649, March.
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