EURO Journal on Computational Optimization

, Volume 7, Issue 2, pp 123–151 | Cite as

Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

  • Vilmar Jefté Rodrigues de Sousa
  • Miguel F. Anjos
  • Sébastien Le DigabelEmail author
Original Paper


We consider the maximum k-cut problem that involves partitioning the vertex set of a graph into k subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We use a cutting-plane algorithm that relies on the early termination of an interior point method, and we study the performance of SDP and linear programming (LP) relaxations for various values of k and instance types. The LP relaxation is strengthened using combinatorial facet-defining inequalities and SDP-based constraints. Our computational results suggest that the LP approach, especially with the addition of SDP-based constraints, outperforms the SDP relaxations for graphs with positive-weight edges and \(k \ge 7\).


Maximum k-cut Graph partitioning Semidefinite programming Eigenvalue constraint Semi-infinite formulation 

Mathematics Subject Classification

65K05 90C22 90C35 



The authors thank the associate editor and two anonymous referees for their helpful comments on an earlier version of this article.


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Copyright information

© The Association of European Operational Research Societies and Springer-Verlag GmbH Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.GERAD and Département de Mathématiques et Génie IndustrielÉcole Polytechnique de MontréalMontrealCanada

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