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Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

  • Miguel F. Anjos
  • Bissan Ghaddar
  • Lena Hupp
  • Frauke Liers
  • Angelika Wiegele

Abstract

This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k=3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that, while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solving the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

Keywords

Valid Inequality Interior Point Method Edge Density Bundle Method Integer Linear Programming Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are grateful to Vera Schmitz for providing us with her implementation of a generic branch-and-bound procedure and to Andreas Schmutzer for help with various aspects of the implementation. We thank Brian Borchers and Christoph Helmberg for support with CSDP and Conic Bundle respectively. We thank an anonymous referee for detailed criticism that helped improve the paper. We also thank Matthias Peinhardt for providing us with data for the instances from [29]. Finally we acknowledge the financial support of the German Science Foundation under contract Li 1675/1 and of the Natural Sciences and Engineering Research Council of Canada.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Miguel F. Anjos
    • 1
  • Bissan Ghaddar
    • 2
  • Lena Hupp
    • 3
  • Frauke Liers
    • 3
  • Angelika Wiegele
    • 4
  1. 1.Canada Research Chair in Discrete Nonlinear Optimization in Engineering, GERADÉcole Polytechnique de MontréalMontréalCanada
  2. 2.Centre for Operational Research and Analysis, Defence Research and Development CanadaDepartment of National DefenceOttawaCanada
  3. 3.Department MathematikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  4. 4.Institut für MathematikAlpen-Adria-Universität KlagenfurtKlagenfurtAustria

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