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A note on the SDP relaxation of the minimum cut problem

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Listed:
  • Hao Hu

    (Clemson University)

  • Xinxin Li

    (Jilin University)

  • Jiageng Wu

    (Jilin University)

Abstract

In a recent paper by Li et al. (Comput Optim Appl 78(3):853–891, 2021), the authors propose an SDP relaxation for solving the minimum cut problem. In this note, we derive this SDP relaxation by Li et al. in a different way which provides a new perspective. Moreover, we give a rigorous proof that this relaxation by Li et al. is a stronger relaxation than the SDP relaxation presented in Pong et al. (Comput Optim Appl 63(2):333–364, 2016). The technique that we use to compare these two relaxations is through establishing a containment relationship between the feasible sets of these two relaxations. This technique is then studied and analyzed in a more general setting. We present some interesting observations about the differences in the optimal sets and values under this technique. Finally, we verify the strength of the SDP relaxation by Li et al. on some random graphs and also on some classical graphs for solving the vertex separator problems in the numerical experiments.

Suggested Citation

  • Hao Hu & Xinxin Li & Jiageng Wu, 2023. "A note on the SDP relaxation of the minimum cut problem," Journal of Global Optimization, Springer, vol. 87(2), pages 857-876, November.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:2:d:10.1007_s10898-022-01235-y
    DOI: 10.1007/s10898-022-01235-y
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    References listed on IDEAS

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    1. Xinxin Li & Ting Kei Pong & Hao Sun & Henry Wolkowicz, 2021. "A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem," Computational Optimization and Applications, Springer, vol. 78(3), pages 853-891, April.
    2. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.
    3. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
    4. 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.
    5. Fanz Rendl & Renata Sotirov, 2018. "The min-cut and vertex separator problem," Computational Optimization and Applications, Springer, vol. 69(1), pages 159-187, January.
    6. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
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