Annals of Operations Research

, Volume 188, Issue 1, pp 155–174 | Cite as

A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem

  • Bissan Ghaddar
  • Miguel F. Anjos
  • Frauke Liers


The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k≥3.


Minimum k-partition Semidefinite programming Branch-and-cut Polyhedral cuts 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Barahona, F., & Mahjoub, A. (1986). On the cut polytope. Mathematical Programming, 36, 157–173. CrossRefGoogle Scholar
  2. Barahona, F., Grötschel, M., Jünger, M., & Reinelt, G. (1988). An application of combinatorial optimization to statistical physics and circuit layout design. Operations Research, 36, 493–513. CrossRefGoogle Scholar
  3. Biq Mac solver. (2007).
  4. Borchers, B. (1999). CSDP, a C library for semidefinite programming. Optimization Methods and Software, 11/12(1–4), 613–623. CrossRefGoogle Scholar
  5. Boros, E., & Hammer, P. (1991). The max-cut problem and quadratic 0–1 optimization: Polyhedral aspects, relaxations and bounds. Annals of Operations Research, 33, 151–180. CrossRefGoogle Scholar
  6. Chopra, S., & Rao, M. R. (1993). The partition problem. Mathematical Programming, 59, 87–115. CrossRefGoogle Scholar
  7. Chopra, S., & Rao, M. R. (1995). Facets of the k-partition problem. Discrete Applied Mathematics, 61, 27–48. CrossRefGoogle Scholar
  8. de Klerk, E., Pasechnik, D., & Warners, J. (2004). Approximate graph colouring and max-k-cut algorithms based on the theta function. Journal of Combinatorial Optimization, 8(3), 267–294. CrossRefGoogle Scholar
  9. Deza, M., & Laurent, M. (1997). Algorithms and combinatorics. Geometry of cuts and metrics. Berlin: Springer. Google Scholar
  10. Deza, M., Grötschel, M., & Laurent, M. (1991). Complete descriptions of small multicut polytopes. Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift, 4, 205–220. Google Scholar
  11. Domingo-Ferrer, J., & Mateo-Sanz, J. M. (2002). Practical data-oriented microaggregation for statistical disclosure control. IEEE Transactions on Knowledge and Data Engineering, 14(1), 189–201. CrossRefGoogle Scholar
  12. Eisenblätter, A. (2002). The semidefinite relaxation of the k-partition polytope is strong. In Proceedings of the 9th international IPCO conference on integer programming and combinatorial optimization (Vol. 2337, pp. 273–290). Google Scholar
  13. Elf, M., Jünger, M., & Rinaldi, G. (2003). Minimizing breaks by maximizing cuts. Operations Research Letters, 31(5), 343–349. CrossRefGoogle Scholar
  14. Frieze, A., & Jerrum, M. (1997). Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18, 67–81. CrossRefGoogle Scholar
  15. Ghaddar, B. (2007). A branch-and-cut algorithm based on semidefinite programming for the minimum k -partition problem. Master’s thesis, University of Waterloo Google Scholar
  16. Goemans, M., & Williamson, D. (1994). New ¾-approximation algorithms for the maximum satisfiability problem. SIAM Journal of Discrete Mathematics, 7(4), 656–666. CrossRefGoogle Scholar
  17. Helmberg, C., & Rendl, F. (1998). Solving quadratic (0, 1)-problems by semidefinite programs and cutting planes. Mathematical Programming, Series A, 82(3), 291–315. CrossRefGoogle Scholar
  18. Kaibel, V., Peinhardt, M., & Pfetsch, M. E. (2007). Orbitopal fixing. In M. Fischetti & D. P. Williamson (Eds.), Lecture notes in computer science : Vol. 4513. IPCO (pp. 74–88). Berlin: Springer. Google Scholar
  19. Lee, L. W., Katzgraber, H. G., & Young, A. P. (2006). Critical behavior of the three- and ten-state short-range Potts glass: A Monte Carlo study. Physical Review B, 74, 104–116. Google Scholar
  20. Liers, F., Jünger, M., Reinelt, G., & Rinaldi, G. (2004). Computing exact ground states of hard ising spin glass problems by branch-and-cut. In New optimization algorithms in physics (pp. 47–68). New York: Wiley. CrossRefGoogle Scholar
  21. Lisser, A., & Rendl, F. (2003). Telecommunication clustering using linear and semidefinite programming. Mathematical Programming, 95, 91–101. CrossRefGoogle Scholar
  22. Lovász, L. (1979). On the Shannon capacity of a graph. IEEE Transactions Information Theory, IT-25, 1–7. CrossRefGoogle Scholar
  23. Mitchell, J. (2001). Branch-and-cut for the k -way equipartition problem (Technical report). Department of Mathematical Sciences, Rensselaer Polytechnic Institute. Google Scholar
  24. Mitchell, J. E. (2003). Realignment in the National Football League: Did they do it right? Naval Research Logistics, 50(7), 683–701. CrossRefGoogle Scholar
  25. Rendl, F., Rinaldi, G., & Wiegele, A. (2007). A branch and bound algorithm for max-cut based on combining semidefinite and polyhedral relaxations. Integer Programming and Combinatorial Optimization, 4513, 295–309. CrossRefGoogle Scholar
  26. Wiegele, A. (2006). Nonlinear optimization techniques applied to combinatorial optimization problems. Ph.D. thesis, Alpen-Adria-Universität Klagenfurt. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Bissan Ghaddar
    • 1
  • Miguel F. Anjos
    • 1
  • Frauke Liers
    • 2
  1. 1.Department of Management SciencesUniversity of WaterlooWaterlooCanada
  2. 2.Institut für InformatikUniversität zu KölnKölnGermany

Personalised recommendations