Advertisement

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
  • 76 Downloads

Abstract

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 \(k \ge 7\).

Keywords

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

Mathematics Subject Classification

65K05 90C22 90C35 

Notes

Acknowledgements

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

References

  1. Ales Z, Knippel A (2016) An extended edge-representative formulation for the k-partitioning problem. Electron Notes Discrete Math 52(Supplement C):333–342 INOC 2015—7th international network optimization conferenceGoogle Scholar
  2. 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 Michael, Reinelt Gerhard (eds) Facets of combinatorial optimization. Springer, Berlin, pp 355–386Google Scholar
  3. Avis D, Umemoto J (2003) Stronger linear programming relaxations of max-cut. Math Program 97(3):451–469Google Scholar
  4. Barahona F, Grötschel M, Jünger M, Reinelt G (1988) An application of combinatorial optimization to statistical physics and circuit layout design. Oper Res 36(3):493–513Google Scholar
  5. Chopra S, Rao MR (1993) The partition problem. Math Program 59(1):87–115Google Scholar
  6. Chopra S, Rao MR (1995) Facets of the k-partition polytope. Discrete Appl Math 61(1):27–48Google Scholar
  7. de Klerk E, Pasechnik DV, Warners JP (2004) On approximate graph colouring and max-\(k\)-cut algorithms based on the \(\theta \)-function. J Comb Optim 8(3):267–294Google Scholar
  8. Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91(2):201–213Google Scholar
  9. Eisenblätter A (2002) The semidefinite relaxation of the \(k\)-partition polytope is strong. In: Cook WJ, Schulz AS (eds) Integer programming and combinatorial optimization. Lecture notes in computer science, vol 237. Springer, Berlin, pp 273–290Google Scholar
  10. Fairbrother J, Letchford AN (2017) Projection results for the k-partition problem. Discrete Optim 26:97–111Google Scholar
  11. Fairbrother J, Letchford AN, Briggs K (2018) A two-level graph partitioning problem arising in mobile wireless communications. Comput Optim Appl 69(3):653–676Google Scholar
  12. Frieze A, Jerrum M (1997) Improved approximation algorithms for maxk-cut and max bisection. Algorithmica 18(1):67–81Google Scholar
  13. Galli L, Kaparis K, Letchford AN (2011) Gap inequalities for non-convex mixed-integer quadratic programs. Oper Res Lett 39(5):297–300Google Scholar
  14. Ghaddar B, Anjos MF, Liers F (2011) A branch-and-cut algorithm based on semidefinite programming for the minimum \(k\)-partition problem. Ann Oper Res 188(1):155–174Google Scholar
  15. Goemans MX, Williamson DP (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J ACM 42(6):1115–1145Google Scholar
  16. Gondzio J (2012) Interior point methods 25 years later. Eur J Oper Res 218:587–601Google Scholar
  17. Gondzio J, González-Brevis P, Munari P (2016) Large-scale optimization with the primal-dual column generation method. Math Program Comput 8(1):47–82Google Scholar
  18. Guennebaud G, Jacob B, et al (2010) Eigen. http://eigen.tuxfamily.org. Accessed Feb 2018
  19. Heggernes P (2006) Minimal triangulations of graphs: a survey. Discrete Math 306(3):297–317Google Scholar
  20. Helmberg C (2000) Semidefinite programming for combinatorial optimization, 1st edn. Konrad-Zuse-Zentrum für Informationstechnik, BerlinGoogle Scholar
  21. Helmberg C, Rendl F (1998) Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math Program 82(3):291–315.  https://doi.org/10.1007/BF01580072 Google Scholar
  22. Hettich R, Kortanek KO (1993) Semi-infinite programming: theory, methods, and applications. SIAM Rev 35(3):380–429Google Scholar
  23. Hopcroft J, Tarjan R (1973) Algorithm 447: efficient algorithms for graph manipulation. Commun ACM 16(6):372–378Google Scholar
  24. Karger D, Motwani R, Sudan M (1998) Approximate graph coloring by semidefinite programming. J ACM 45(2):246–265Google Scholar
  25. Krishnan K, Mitchell JE (2001) Semi-infinite linear programming approaches to semidefinite programming problems. Technical Report 37, Fields Institute Communications SeriesGoogle Scholar
  26. Krishnan K, Mitchell JE (2006) A semidefinite programming based polyhedral cut and price approach for the maxcut problem. Comput Optim Appl 33(1):51–71Google Scholar
  27. Krislock N, Malick J, Roupin F (2012) Improved semidefinite bounding procedure for solving max-cut problems to optimality. Math Program 143(1):61–86Google Scholar
  28. Laurent M, Poljak S (1996) Gap inequalities for the cut polytope. Eur J Comb 17(2):233–254Google Scholar
  29. Liers F, Jünger M, Reinelt G, Rinaldi G (2005) Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: New optimization algorithms in physics. Wiley-VCH Verlag GmbH & Co. KGaA, pp 47–69Google Scholar
  30. Lisser A, Rendl F (2003) Graph partitioning using linear and semidefinite programming. Math Program 95(1):91–101Google Scholar
  31. Ma F, Hao J-K (2017) A multiple search operator heuristic for the max-k-cut problem. Ann Oper Res 248(1):365–403Google Scholar
  32. Mitchell JE (2000) Computational experience with an interior point cutting plane algorithm. SIAM J Optim 10(4):1212–1227Google Scholar
  33. Mitchell JE (2003) Realignment in the National Football League: did they do it right? Naval Res Logist 50(7):683–701Google Scholar
  34. Mitchell JE, Pardalos PM, Resende MGC (1999) Interior point methods for combinatorial optimization. In: Du D-Z, Pardalos PM (eds) Handbook of combinatorial optimization, vol 1–3. Springer, Boston, pp 189–297Google Scholar
  35. Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100Google Scholar
  36. Moré JJ, Wild SM (2009) Benchmarking derivative-free optimization algorithms. SIAM J Optim 20(1):172–191Google Scholar
  37. Mosek ApS (2015) MOSEK http://www.mosek.com. Accessed Feb 2018
  38. Munari P, Gondzio J (2013) Using the primal-dual interior point algorithm within the branch-price-and-cut method. Comput Oper Res 40(8):2026–2036Google Scholar
  39. Niu C, Li Y, Qingyang Hu R, Ye F (2017) Femtocell-enhanced multi-target spectrum allocation strategy in LTE-A HetNets. IET Commun 11(6):887–896Google Scholar
  40. Palagi L, Piccialli V, Rendl F, Rinaldi G, Wiegele A (2011) Computational approaches to max-cut. In: Anjos MF, Lasserre JB (eds) Handbook of semidefinite, conic and polynomial optimization: theory, algorithms, software and applications. International series in operations research and management science. Springer, New York, pp 821–847Google Scholar
  41. Papadimitriou CH, Yannakakis M (1991) Optimization, approximation, and complexity classes. J Comput Syst Sci 43(3):425–440Google Scholar
  42. Rendl F (2012) Semidefinite relaxations for partitioning, assignment and ordering problems. 4OR 10(4):321–346Google Scholar
  43. Rendl F, Rinaldi G, Wiegele A (2010) Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math Program 121(2):307–335Google Scholar
  44. Rinaldi G (2018) Rudy, a graph generator. https://www-user.tu-chemnitz.de/~helmberg/sdp_software.html. Accessed Feb 2018
  45. Rodrigues de Sousa VJ, Anjos MF, Le Digabel S (2018) Computational study of valid inequalities for the maximum \(k\)-cut problem. Ann Oper Res 265(1):5–27.  https://doi.org/10.1007/s10479-017-2448-9 Google Scholar
  46. Seidman SB (1983) Network structure and minimum degree. Soc Netw 5(3):269–287Google Scholar
  47. Sherali HD, Fraticelli BMP (2002) Enhancing RLT relaxations via a new class of semidefinite cuts. J Glob Optim 22(1–4):233–261Google Scholar
  48. Shor N Z (1998) Semidefinite programming bounds for extremal graph problems. Springer, Boston, pp 265–298Google Scholar
  49. Wang G, Hijazi H (2017) Exploiting sparsity for the min k-partition problem. arXiv e-prints. arXiv:1709.00485
  50. Wiegele A (2015) Biq mac library—binary quadratic and max cut library. http://biqmac.uni-klu.ac.at/biqmaclib.html. Accessed Feb 2018

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

Personalised recommendations