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

Sheng-Jie Chen (), Liang Chen (), Guang-Ming Li () and Yu-Hong Dai ()
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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
Liang Chen: Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, 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

Computational Optimization and Applications, 2025, vol. 92, issue 1, No 7, 215-263

Abstract: 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.

Keywords: Critical nodes detection; Integer programming; Preprocessing; Polyhedral combinatorics; Valid inequality; Separation (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10589-025-00698-5

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