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

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Facets of Combinatorial Optimization

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.

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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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Authors and Affiliations

  1. Canada Research Chair in Discrete Nonlinear Optimization in Engineering, GERAD, École Polytechnique de Montréal, Montréal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. Centre for Operational Research and Analysis, Defence Research and Development Canada, Department of National Defence, 101 Colonel By Drive, Ottawa, ON, Canada, K1A 0K2

    Bissan Ghaddar

  3. Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Lena Hupp & Frauke Liers

  4. Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstr. 65-67, 9020, Klagenfurt, Austria

    Angelika Wiegele

Authors
  1. Miguel F. Anjos
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  2. Bissan Ghaddar
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  3. Lena Hupp
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  4. Frauke Liers
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  5. Angelika Wiegele
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Corresponding author

Correspondence to Frauke Liers .

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Editors and Affiliations

  1. Dept. of Computer Science, University of Cologne, Pohligstr. 1, Cologne, 50969, Germany

    Michael Jünger

  2. Dept. of Computer Science, University of Heidelberg, Im Neuenheimer Feld 326, Heidelberg, 69120, Germany

    Gerhard Reinelt

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Anjos, M.F., Ghaddar, B., Hupp, L., Liers, F., Wiegele, A. (2013). Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method. In: Jünger, M., Reinelt, G. (eds) Facets of Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38189-8_15

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