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A note on the SDP relaxation of the minimum cut problem

Author

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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