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A branch-and-bound algorithm for solving max- k -cut problem
Journal of Global Optimization ( IF 1.8 ) Pub Date : 2021-02-11 , DOI: 10.1007/s10898-021-00999-z
Cheng Lu , Zhibin Deng

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.



中文翻译:

解决最大k割问题的分支定界算法

最大k割问题是最著名的组合优化问题之一。在本文中,我们设计了一种有效的分支定界算法来解决最大k割问题。我们提出了一种半定松弛,它与文献中的常规半定松弛一样紧密,但是更适合作为分支定界框架中的松弛方法。然后,我们开发了一种分支定界算法,该算法利用了所提出的半定松弛的独特结构,并应用了一种与现有算法中常用的分支方法不同的分支方法。max- k解集的对称结构讨论了割问题,并设计了一种策略来减少枚举过程中子问题的冗余。数值结果表明,该算法是可行的。在边缘超过350个的实例上,它的性能比Gurobi好,并且在某些类型的测试实例上,它的效率比最新的算法bundleBC高。

更新日期:2021-02-11
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