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 LiersEmail author
  • Angelika Wiegele


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.


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.



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.


  1. 1.
    Anjos, M.F., Wolkowicz, H.: Geometry of semidefinite max-cut relaxations via matrix ranks. J. Comb. Optim. 6(3), 237–270 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Anjos, M.F., Wolkowicz, H.: Strengthened semidefinite relaxations via a second lifting for the max-cut problem. Discrete Appl. Math. 119(1–2), 79–106 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Anjos, M.F., Liers, F., Pardella, G., Schmutzer, A.: Engineering branch-and-cut algorithms for the equicut problem. Cahier du GERAD G-2012-15, GERAD, Montreal, QC, Canada (2012). In: Fields Institute Communications on Discrete Geometry and Optimization. Springer, Berlin (2013, to appear) Google Scholar
  4. 4.
    Armbruster, M., Fügenschuh, M., Helmberg, C., Martin, A.L.: LP and SDP branch-and-cut algorithms for the minimum graph bisection problem: a computational comparison. Math. Program. Comput. 4(3), 275–306 (2012) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barahona, F., Mahjoub, A.: On the cut polytope. Math. Program. 36, 157–173 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36, 493–513 (1988) zbMATHCrossRefGoogle Scholar
  7. 7.
    BiqMac solver. Accessed 07 June 2012
  8. 8.
    Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11/12(1–4), 613–623 (1999) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Boros, E., Hammer, P.: The max-cut problem and quadratic 0–1 optimization: polyhedral aspects, relaxations and bounds. Ann. Oper. Res. 33, 151–180 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brunetta, L., Conforti, M., Rinaldi, G.: A branch-and-cut algorithm for the equicut problem. Math. Program., Ser. B 78(2), 243–263 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chopra, S., Rao, M.R.: The partition problem. Math. Program. 59, 87–115 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chopra, S., Rao, M.R.: Facets of the k-partition problem. Discrete Appl. Math. 61, 27–48 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Conic Bundle Library. Accessed 28 October 2011
  14. 14.
    de Klerk, E., Pasechnik, D., Warners, J.: On approximate graph colouring and max-k-cut algorithms based on the ϑ-function. J. Comb. Optim. 8(3), 267–294 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics. Springer, Berlin (1997) zbMATHGoogle Scholar
  16. 16.
    Deza, M., Grötschel, M., Laurent, M.: Complete descriptions of small multicut polytopes. In: Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift, pp. 205–220, Am. Math. Soc., Providence (1991) Google Scholar
  17. 17.
    Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program., Ser. A, 91(2), 201–213 (2002) zbMATHCrossRefGoogle Scholar
  18. 18.
    Eisenblätter, A.: The semidefinite relaxation of the k-partition polytope is strong. In: Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 2337, pp. 273–290. Springer, Berlin (2002) CrossRefGoogle Scholar
  19. 19.
    Elf, M., Jünger, M., Rinaldi, G.: Minimizing breaks by maximizing cuts. Oper. Res. Lett. 31(5), 343–349 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fischer, I., Gruber, G., Rendl, F., Sotirov, R.: Computational experience with a bundle approach for semidefinite cutting plane relaxations of max-cut and equipartition. Math. Program., Ser. B 105(2–3), 451–469 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Frieze, A., Jerrum, M.: Improved approximation algorithms for max k-cut and max bisection. Algorithmica 18, 67–81 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem. Ann. Oper. Res. 188(1), 155–174 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Goemans, M., Williamson, D.: New \(\frac{3}{4}\)-approximation algorithms for the maximum satisfiability problem. SIAM J. Discrete Math. 7(4), 656–666 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. In: The Sharpest Cut. MPS/SIAM Ser. Optim., pp. 233–256. SIAM, Philadelphia (2004) Google Scholar
  25. 25.
    Helmberg, C., Kiwiel, K.C.: A spectral bundle method with bounds. Math. Program., Ser. A 93(2), 173–194 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program., Ser. A 82(3), 291–315 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000) (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6(2), 342–361 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kaibel, V., Peinhardt, M., Pfetsch, M.: Orbitopal fixing. In: Fischetti, M., Williamson, D. (eds.) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 4513, pp. 74–88. Springer, Berlin (2007) CrossRefGoogle Scholar
  30. 30.
    Kaibel, V., Peinhardt, M., Pfetsch, M.: Orbitopal fixing. Discrete Optim. 8(4), 595–610 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985) zbMATHGoogle Scholar
  32. 32.
    Laurent, M.: Semidefinite relaxations for max-cut. In: The Sharpest Cut. MPS/SIAM Ser. Optim., pp. 257–290. SIAM, Philadelphia (2004) Google Scholar
  33. 33.
    Laurent, M., Poljak, S.: On a positive semidefinite relaxation of the cut polytope. Linear Algebra Appl. 223/224, 439–461 (1995) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Laurent, M., Poljak, S.: On the facial structure of the set of correlation matrices. SIAM J. Matrix Anal. Appl. 17(3), 530–547 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Lemaréchal, C.: Bundle methods in nonsmooth optimization. In: Nonsmooth Optimization, Proc. IIASA Workshop, Laxenburg, 1977. IIASA Proc. Ser., vol. 3, pp. 79–102. Pergamon, Oxford (1978) Google Scholar
  36. 36.
    Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program., Ser. B 69(1), 111–147 (1995) zbMATHCrossRefGoogle Scholar
  37. 37.
    Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: New Optimization Algorithms in Physics, pp. 47–68. Wiley, New York (2004) Google Scholar
  38. 38.
    Liers, F., Lukic, J., Marinari, E., Pelissetto, A., Vicari, E.: Zero-temperature behavior of the random-anisotropy model in the strong-anisotropy limit. Phys. Rev. B 76(17), 174423 (2007) CrossRefGoogle Scholar
  39. 39.
    Lisser, A., Rendl, F.: Telecommunication clustering using linear and semidefinite programming. Math. Program. 95, 91–101 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Margot, F.: Pruning by isomorphism in branch-and-cut. Math. Program., Ser. A, 94(1), 71–90 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Margot, F.: Exploiting orbits in symmetric ILP. Math. Program., Ser. B, 98(1–3), 3–21 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Max-k-cut instances. Accessed 10 March 2011
  43. 43.
    Mitchell, J.: Branch-and-cut for the k-way equipartition problem. Technical report, Department of Mathematical Sciences, Rensselaer Polytechnic Institute (2001) Google Scholar
  44. 44.
    Mitchell, J.E.: Realignment in the National Football League: did they do it right? Nav. Res. Logist. 50(7), 683–701 (2003) zbMATHCrossRefGoogle Scholar
  45. 45.
    Palagi, L., Piccialli, V., Rendl, F., Rinaldi, G., Wiegele, A.: Computational approaches to max-cut. In: Handbook on Semidefinite, Conic and Polynomial Optimization. Internat. Ser. Oper. Res. Management Sci., vol. 166, pp. 821–847. Springer, New York (2012) CrossRefGoogle Scholar
  46. 46.
    Poljak, S., Rendl, F.: Solving the max-cut problem using eigenvalues. Discrete Appl. Math. 62(1–3), 249–278 (1995). doi: 10.1016/0166-218X(94)00155-7 MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307–335 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Rinaldi, G.: Rudy. Accessed 07 April 2010
  49. 49.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Spin-glass server. Accessed 07 June 2012

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Miguel F. Anjos
    • 1
  • Bissan Ghaddar
    • 2
  • Lena Hupp
    • 3
  • Frauke Liers
    • 3
    Email author
  • 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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