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Spin glass approach to the feedback vertex set problem

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  • Hai-Jun Zhou

Abstract

A feedback vertex set (FVS) of an undirected graph is a set of vertices that contains at least one vertex of each cycle of the graph. The feedback vertex set problem consists of constructing a FVS of size less than a certain given value. This combinatorial optimization problem has many practical applications, but it is in the nondeterministic polynomial-complete class of worst-case computational complexity. In this paper we define a spin glass model for the FVS problem and then study this model on the ensemble of finite-connectivity random graphs. In our model the global cycle constraints are represented through the local constraints on all the edges of the graph, and they are then treated by distributed message-passing procedures such as belief propagation. Our belief propagation-guided decimation algorithm can construct nearly optimal feedback vertex sets for single random graph instances and regular lattices. We also design a spin glass model for the FVS problem on a directed graph. Our work will be very useful for identifying the set of vertices that contribute most significantly to the dynamical complexity of a large networked system. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Hai-Jun Zhou, 2013. "Spin glass approach to the feedback vertex set problem," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 86(11), pages 1-9, November.
    • Handle: RePEc:spr:eurphb:v:86:y:2013:i:11:p:1-9:10.1140/epjb/e2013-40690-1
    DOI: 10.1140/epjb/e2013-40690-1
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    Cited by:

    1. Xiao-Long Ren & Niels Gleinig & Dijana Tolić & Nino Antulov-Fantulin, 2018. "Underestimated Cost of Targeted Attacks on Complex Networks," Complexity, Hindawi, vol. 2018, pages 1-15, January.
    2. Wandelt, Sebastian & Lin, Wei & Sun, Xiaoqian & Zanin, Massimiliano, 2022. "From random failures to targeted attacks in network dismantling," Reliability Engineering and System Safety, Elsevier, vol. 218(PA).
    3. Li Zeng & Changjun Fan & Chao Chen, 2023. "Leveraging Minimum Nodes for Optimum Key Player Identification in Complex Networks: A Deep Reinforcement Learning Strategy with Structured Reward Shaping," Mathematics, MDPI, vol. 11(17), pages 1-13, August.
    4. Feng, Zhidan & Song, Huimin & Qi, Xingqin, 2024. "A novel algorithm for the generalized network dismantling problem based on dynamic programming," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).

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    Keywords

    Statistical and Nonlinear Physics;

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