Max-k-Cut by the Discrete Dynamic Convexized Method

  • Authors:

  • Wenxing Zhu;Geng Lin;M. M. Ali

  • Affiliations:

  • Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350108, China;Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350108, China;School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa

  • Venue:
  • INFORMS Journal on Computing
  • Year:
  • 2013

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Abstract

In this paper, we propose a “multistart-type” algorithm for solving the max-k-cut problem. Central to our algorithm is an auxiliary function we propose. We formulate the max-k-cut problem as an explicit mathematical form, which allows us to use an easy implementable local search. The construction of the auxiliary function requires a local maximizer of the max-k-cut problem. If the best local maximizer obtained is used in the construction of the auxiliary function, then the local maximization of the auxiliary function leads to a better maximizer of the max-k-cut problem. This proves to be a good strategy to escape from the current local optima and to search a broader solution space. Indeed, we have shown, both numerically and theoretically, that the maximization of the auxiliary function by the local search method can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. Computational results on many test instances with different sizes and densities show that the proposed algorithm is efficient and stable to find approximate global solutions for the max-k-cut problems. Although we have presented results for k ≥ 2, the robustness of our algorithm is shown for k = 2 by comparisons with a number of recent methods. A number of theoretical results are also presented, which justify the design of our algorithm.