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A branch-and-bound algorithm for solving max-k-cut problem

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Abstract

The max-k-cut problem is one of the most well-known combinatorial optimization problems. In this paper, we design an efficient branch-and-bound algorithm to solve the max-k-cut problem. We propose a semidefinite relaxation that is as tight as the conventional semidefinite relaxations in the literature, but is more suitable as a relaxation method in a branch-and-bound framework. We then develop a branch-and-bound algorithm that exploits the unique structure of the proposed semidefinite relaxation, and applies a branching method different from the one commonly used in the existing algorithms. The symmetric structure of the solution set of the max-k-cut problem is discussed, and a strategy is devised to reduce the redundancy of subproblems in the enumeration procedure. The numerical results show that the proposed algorithm is promising. It performs better than Gurobi on instances that have more than 350 edges, and is more efficient than the state-of-the-art algorithm bundleBC on certain types of test instances.

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Notes

  1. The test instances are arisen from applications in statistical physics, and available at http://biqmac.uni-klu.ac.at/library/mac/ising.

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Acknowledgements

The authors would like to thank the editors and two anonymous reviewers for their valuable suggestions and comments, which have helped to improve the quality of this paper significantly. The authors would especially thank Dr. Vilmar de Sousa for sharing the 62 test instances used in the numerical experiments of this paper.

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Correspondence to Zhibin Deng.

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Lu’s research has been supported by National Natural Science Foundation of China Grant Nos. 11701177 and 11771243, and Fundamental Research Funds for the Central Universities Grant No. 2018ZD14. Deng’s research has been supported by University of Chinese Academy of Sciences.

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Lu, C., Deng, Z. A branch-and-bound algorithm for solving max-k-cut problem. J Glob Optim 81, 367–389 (2021). https://doi.org/10.1007/s10898-021-00999-z

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