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
The test instances are arisen from applications in statistical physics, and available at http://biqmac.uni-klu.ac.at/library/mac/ising.
References
Anjos, M.F., Wolkowicz, H.: Strengthened semidefinite relaxations via a second lifting for the max-cut problem. Discrete Appl. Math. 119, 79–106 (2002)
Anjos, M.F., Ghaddar, B., Hupp, L., Liers, F., Wiegele, A.: Solving \(k\)-way graph partitioning problems to optimality: the impact of semidefinite relaxations and the bundle method. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 355–386. Springer, Berlin (2013)
Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programs: a review and comparisons. Math. Program. 129, 129–157 (2011)
Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36, 493–513 (1988)
Buchheim, C., Montenegro, M., Wiegele, A.: A coordinate ascent method for solving semidefinite relaxations of non-convex quadratic integer programs. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds.) ISCO. Lecture Notes in Computer Science, vol. 9849, pp. 110–122. Springer, Berlin (2016)
Buchheim, C., Montenegro, M., Wiegele, A.: SDP-based branch-and-bound for non-convex quadratic integer optimization. J. Global Optim. 73, 485–514 (2019)
Chopra, S., Rao, M.R.: The partition problem. Math. Program. 59, 87–115 (1993)
Chopra, S., Rao, M.R.: Facets of the \(k\)-partition problem. Discrete Appl. Math. 61, 27–48 (1995)
de Klerk, E., Pasechnik, D., Warners, J.: On approximate graph colouring and max-\(k\)-cut algorithms based on the \(\vartheta \)-function. J. Comb. Optim. 8, 267–294 (2004)
de Sousa, V.J.R., Anjos, M.F., Digabel, S.L.: Computational study of a branching algorithm for the maximum \(k\)-cut problem. http://www.optimization-online.org/DB_HTML/2020/02/7629.html (2020)
de Sousa, V.J.R., Anjos, M.F., Digabel, S.L.: Computational study of valid inequalities for the maximum \(k\)-cut problem. Ann. Oper. Res. 265, 5–27 (2018)
Domingo-Ferrer, J., Mateo-Sanz, J.M.: Practical data-oriented microaggregation for statistical disclosure control. IEEE Trans. Knowl. Data Eng. 14, 189–201 (2002)
Eisenblätter, A.: The semidefinite relaxation of the \(k\)-partition polytope is strong. In: Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 2337, pp. 273–290. Springer, Berlin (2002)
Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and equipartition. Math. Program. 105, 451–469 (2006)
Frieze, A., Jerrum, M.: Improved approximation algorithms for max \(k\)-cut and max bisection. Algorithmica 18, 67–81 (1997)
Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum \(k\)-partition problem. Ann. Oper. Res. 188, 155–174 (2011)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)
Goemans, M.X., Williamson, D.P.: Approximation algorithms for Max-3-Cut and other problems via complex semidefinite programming. J. Comput. Syst. Sci. 68, 442–470 (2004)
Jarre, F., Lieder, F., Liu, Y.-F., Lu, C.: Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting. J. Global Optim. 76, 913–932 (2020)
Krislock, N., Malick, J., Roupin, F.: Improved semidefinite bounding procedure for solving Max-Cut problems to optimality. Math. Program. 143, 62–86 (2014)
Luo, Z.-Q., Ma, W.K., So, A.M.C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems: from its practical deployments and scope of applicability to key theoretical results. IEEE Signal. Proc. Mag. 27, 20–34 (2010)
Mosek ApS. mosek. http://www.mosek.com (2020)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)
Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307–335 (2010)
Rinaldi, G.: Rudy. http://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz (1998)
Sotirov, R.: An efficient semidefinite programming relaxation for the graph partition problem. INFORMS J. Comput. 26, 16–30 (2014)
Wang, G., Hijazi, H.: Exploiting sparsity for the min \(k\)-partition problem. Math. Program. Comput. 12, 109–130 (2020)
Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203–230 (2010)
Wolkowicz, H., Zhao, Q.: Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97, 461–479 (1999)
Zhao, X.Y., Sun, D., Toh, K.C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)
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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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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DOI: https://doi.org/10.1007/s10898-021-00999-z