Annals of Operations Research

, Volume 265, Issue 1, pp 5–27 | Cite as

Computational study of valid inequalities for the maximum k-cut problem

  • Vilmar Jefté Rodrigues de SousaEmail author
  • Miguel F. Anjos
  • Sébastien Le Digabel
Paths, Pivots, and Practice: The Power of Optimization


We consider the maximum k-cut problem that consists in partitioning the vertex set of a graph into k subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study of four classes of inequalities from the literature: clique, general clique, wheel and bicycle wheel. We considered 10 combinations of these classes and tested them on both dense and sparse instances for \( k \in \{3,4,5,7\} \). Our computational results suggest that the bicycle wheel and wheel are the strongest inequalities for \( k=3 \), and that for \( k \in \{4,5,7\} \) the wheel inequalities are the strongest by far. Furthermore, we observe an improvement in the performance for all choices of k when both bicycle wheel and wheel are used, at the cost of 72% more CPU time on average when compared with using only one of them.


Maximum k-cut Graph partitioning Semidefinite programming Computational study 

Mathematics Subject Classification

65K05 90C22 90C35 



This research was supported by Discovery Grants 312125 (M.F. Anjos) and 418250 (S. Le Digabel) from the Natural Sciences and Engineering Research Council of Canada. Moreover, we thank three anonymous reviewers who helped us to significantly improve this paper.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Vilmar Jefté Rodrigues de Sousa
    • 1
    Email author
  • Miguel F. Anjos
    • 2
  • Sébastien Le Digabel
    • 1
  1. 1.GERAD and Département de Mathématiques et Génie IndustrielPolytechnique MontréalMontrealCanada
  2. 2.Canada Research Chair in Discrete Nonlinear Optimization in Engineering, GERAD and Département de Mathématiques et Génie IndustrielPolytechnique MontréalMontrealCanada

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