Gap Inequalities for the Max-Cut Problem: A Cutting-Plane Algorithm

  • Laura Galli
  • Konstantinos Kaparis
  • Adam N. Letchford
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

Laurent & Poljak introduced a class of valid inequalities for the max-cut problem, called gap inequalities, which include many other known inequalities as special cases. The gap inequalities have received little attention and are poorly understood. This paper presents the first ever computational results. In particular, we describe heuristic separation algorithms for gap inequalities and their special cases, and show that an LP-based cutting-plane algorithm based on these separation heuristics can yield very good upper bounds in practice.

Keywords

Triangle Inequality Valid Inequality Primal Stabilisation Linear Programming Relaxation Separation Algorithm 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avis, D.: On the Complexity of Testing Hypermetric, Negative Type, k-Gonal and Gap Inequalities. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 51–59. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Avis, D., Umemoto, J.: Stronger linear programming relaxations of max-cut. Math. Program. 97, 451–469 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barahona, F., Mahjoub, A.R.: On the cut polytope. Math. Program. 36, 157–173 (1986)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Meth. & Software 11, 613–623 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. Wiley, New York (1998)MATHGoogle Scholar
  6. 6.
    Deza, M.: On the Hamming geometry of unitary cubes. Soviet Physics Doklady 5, 940–943 (1961)Google Scholar
  7. 7.
    Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Springer, Berlin (1997)MATHGoogle Scholar
  8. 8.
    Galli, L., Letchford, A.N.: Small bipartite subgraph polytopes. Oper. Res. Lett. 38(5), 337–340 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Galli, L., Kaparis, K., Letchford, A.N.: Gap inequalities for non-convex mixed-integer quadratic programs. Oper. Res. Lett. 39(5), 297–300 (2011)MathSciNetMATHGoogle Scholar
  10. 10.
    Galli, L., Kaparis, K., Letchford, A.N.: Complexity results for the gap inequalities for the max-cut problem. To appear in Oper. Res. Lett (2011)Google Scholar
  11. 11.
    Giandomenico, M., Letchford, A.N.: Exploring the relationship between max-cut and stable set relaxations. Math. Program. 106, 159–175 (2006)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Ass. Comp. Mach. 42, 1115–1145 (1995)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Helmberg, C., Rendl, F.: Solving quadratic (0,1)-programs by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998)MathSciNetMATHGoogle Scholar
  14. 14.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  15. 15.
    Kelly, J.B.: Hypermetric spaces. In: The Geometry of Metric and Linear Spaces. Lecture Notes in Mathematics, vol. 490, pp. 17–31. Springer, Berlin (1974)CrossRefGoogle Scholar
  16. 16.
    Laurent, M.: Max-cut problem. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliographies in Combinatorial Optimization, pp. 241–259. Wiley, Chichester (1997)Google Scholar
  17. 17.
    Laurent, M., Poljak, S.: On a positive semidefinite relaxation of the cut polytope. Lin. Alg. Appl. 223/224, 439–461 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Laurent, M., Poljak, S.: Gap inequalities for the cut polytope. SIAM Journal on Matrix Analysis 17, 530–547 (1996)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Letchford, A.N., Sørensen, M.M.: Binary Positive Semidefinite Matrices and Associated Integer Polytopes. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 125–139. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Marsten, R.E., Hogan, W.W., Blankenship, J.W.: The BOXSTEP method for large-scale optimization. Oper. Res. 23, 389–405 (1975)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Poljak, S., Rendl, F.: Nonpolyhedral relaxations of graph bisection problems. SIAM J. on Opt. 5, 467–487 (1995)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. In: Cook, W., Lovász, L., Seymour, P. (eds.) Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 20, pp. 181–244. American Mathematical Society (1995)Google Scholar
  23. 23.
    Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. IMA Volumes in Mathematics and its Applications, vol. 154, pp. 407–426. Springer, Berlin (2012)CrossRefGoogle Scholar
  24. 24.
    Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44, 522–536 (1938)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Laura Galli
    • 1
  • Konstantinos Kaparis
    • 2
  • Adam N. Letchford
    • 2
  1. 1.Warwick Business SchoolUniversity of WarwickUnited Kingdom
  2. 2.Department of Management ScienceLancaster UniversityUnited Kingdom

Personalised recommendations