Mathematical Programming

, Volume 146, Issue 1–2, pp 351–378 | Cite as

Lifting and separation procedures for the cut polytope

  • Thorsten Bonato
  • Michael Jünger
  • Gerhard Reinelt
  • Giovanni Rinaldi
Full Length Paper Series A

Abstract

The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, in comparison, only little research has been conducted for the cut polytope on arbitrary graphs, in particular separation algorithms have received only little attention. In this study we describe new separation and lifting procedures for the cut polytope on general graphs. These procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid, and sometimes facet defining, inequalities for the cut polytope on arbitrary graphs in a cutting plane framework. We report computational results on a set of well-established benchmark problems.

Keywords

Max-cut problem Cut polytope Separation algorithm  Branch-and-cut Unconstrained boolean quadratic programming   Ising spin glass model 

Mathematics Subject Classification (2000)

90C27 90C57 90C20 90C09 82D30 

References

  1. 1.
    Anjos, M., Lasserre, J. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization. Springer, Berlin (2012)MATHGoogle Scholar
  2. 2.
    Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)Google Scholar
  3. 3.
    Avis, D., Imai, H., Ito, T., Sasaki, Y.: Two-party Bell inequalities derived from combinatorics via triangular elimination. J. Phys. A 38, 10971–10987 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Avis, D., Imai, H., Ito, T.: Generating facets for the cut polytope of a graph by triangular elimination. Math. Program. 112, 303–325 (2008)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Avis, D., Ito, T.: New classes of facets for the cut polytope and tightness of \(I_{mm22}\) Bell inequalities. Discrete Appl. Math. 155, 1689–1699 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Barahona, F.: On cuts and matchings in planar graphs. Math. Program. 60, 53–68 (1993)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Barahona, F., Mahjoub, A.R.: On the cut polytope. Math. Program. 36, 157–173 (1986)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefMATHGoogle Scholar
  9. 9.
    Boros, E., Hammer, P.L.: Cut-polytopes, boolean quadric polytopes and nonnegative quadratic pseudo-boolean functions. Math. Oper. Res. 18, 245–253 (1993)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Buchheim, C., Liers, F., Oswald, M.: Local cuts revisited. Oper. Res. Lett. 36, 430–433 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cheng, E.: Separating subdivision of bicycle wheel inequalities over cut polytopes. Oper. Res. Lett. 23, 13–19 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    De Simone, C.: Lifting facets of the cut polytope. Oper. Res. Lett. 9, 341–344 (1990)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    De Simone, C., Rinaldi, G.: A cutting plane algorithm for the max-cut problem. Optim. Methods Softw. 3, 195–214 (1994)CrossRefGoogle Scholar
  14. 14.
    De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of Ising spin glasses: new experimental results with a branch and cut algorithm. J. Stat. Phys. 80, 487–496 (1995)CrossRefMATHGoogle Scholar
  15. 15.
    De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of two-dimensional \(\pm J\) Ising spin glasses. J. Stat. Phys. 84, 1363–1371 (1996)CrossRefMATHGoogle Scholar
  16. 16.
    Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)CrossRefGoogle Scholar
  17. 17.
    Gerards, A.M.H.: Testing the odd bicycle wheel inequalities for the bipartite subgraph polytope. Math. Oper. Res. 10, 359–360 (1985)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Jünger, M., Thienel, S.: The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Softw Pract Exp 30, 1325–1352 (2000)CrossRefMATHGoogle Scholar
  21. 21.
    Laurent, M., Poljak, S.: One-third-integrality in the max-cut problem. Math. Program. 71, 29–50 (1995)MATHMathSciNetGoogle Scholar
  22. 22.
    Liers, F.: Contributions to Determining Exact Ground-States of Ising Spin-Glasses and to their Physics. PhD Thesis, University of Cologne (2004)Google Scholar
  23. 23.
    Liers, F., Jünger, M., Reinelt, G., Rinaldi, G.: Computing exact ground states of hard Ising spin glass problems by branch-and-cut. In: Hartmann, A., Rieger, H. (eds.) New Optim. Algorithms Phys., pp. 47–70. Wiley-VCH, London (2004)Google Scholar
  24. 24.
    Mannino, C.: Personal communication (2011)Google Scholar
  25. 25.
    Marsaglia, G., Bray, T.A.: A convenient method for generating normal variables. SIAM Rev. 6, 260–264 (1964)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Poljak, S., Tuza, Z.: Maximum cuts and large bipartite subgraphs. In: Cook, W. et al. (eds.) Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 20, pp. 181–244 (1995)Google Scholar
  27. 27.
    Rendl, F., et al.: Semidefinite relaxations for integer programming. In: Jünger, M. et al. (eds.) 50 years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys, pp. 687–726. Springer, Berlin (2010)Google Scholar
  28. 28.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121, 307–335 (2010)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Rinaldi, G.: Rudy: a graph generator. http://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz (1998)
  30. 30.
    Wiegele, A.: BiqMac library. biqmac.uni-klu.ac.at/biqmaclib.html (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  • Thorsten Bonato
    • 1
  • Michael Jünger
    • 2
  • Gerhard Reinelt
    • 1
  • Giovanni Rinaldi
    • 3
  1. 1.Institut für InformatikUniversität HeidelbergHeidelbergGermany
  2. 2.Institut für InformatikUniversität zu KölnCologneGermany
  3. 3.Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, CNRRomeItaly

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