Improving Cutting Plane Generation with 0-1 Inequalities by Bi-criteria Separation

  • Edoardo Amaldi
  • Stefano Coniglio
  • Stefano Gualandi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6049)

Abstract

In cutting plane-based methods, the question of how to generate the “best possible” cuts is a central and critical issue. We propose a bi-criteria separation problem for generating valid inequalities that simultaneously maximizes the cut violation and a measure of the diversity between the new cut and the previously generated cut(s). We focus on problems with cuts having 0-1 coefficients, and use the 1-norm as diversity measure. In this context, the bi-criteria separation amounts to solving the standard single-criterion separation problem (maximizing violation) with different coefficients in the objective function. We assess the impact of this general approach on two challenging combinatorial optimization problems, namely the Min Steiner Tree problem and the Max Clique problem. Computational experiments show that the cuts generated by the bi-criteria separation are much stronger than those obtained by just maximizing the cut violation, and allow to close a larger fraction of the gap in a smaller amount of time.

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References

  1. 1.
    Achterberg, T.: Constrained Integer Programming. PhD thesis, Technische Universitat at Berlin (2007)Google Scholar
  2. 2.
    Andreello, G., Caprara, A., Fischetti, M.: Embedding 0,1/2-cuts in a branch-and-cut framework: a computational study. J. on Computing 19(2), 229–238 (2007)MathSciNetGoogle Scholar
  3. 3.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58(1-3), 295–324 (1993)CrossRefGoogle Scholar
  4. 4.
    Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0–1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42(9), 1229–1246 (1996)MATHCrossRefGoogle Scholar
  5. 5.
    Balas, E., Fischetti, M., Zanette, A.: Can pure cutting plane algorithms work? In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 416–434. Springer, Heidelberg (2008)Google Scholar
  6. 6.
    Fischetti, M., Lodi, A.: Optimizing over the first Chvatal closure. Math. Program. 110(1), 3–20 (2006)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Johnson, D.S., Trick, M.A.: Cliques, Coloring, and Satisfiability. In: Second DIMACS Implementation Challenge. DIMACS Series in Disc. Math. and Theo. Comp. Scie. Amer. Math. Soc., vol. 26 (1996)Google Scholar
  8. 8.
    Koch, T., Martin, A.: Solving Steiner Tree Problems in Graphs to Optimality. Networks 32(3), 207–232 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Koch, T., Martin, A., Voß, S.: SteinLib: An updated library on steiner tree problems in graphs. Technical Report ZIB-Report 00-37, ZIB Berlin, Takustr. 7, Berlin (2000)Google Scholar
  10. 10.
    Marchand, H., Martin, A., Weismantel, R., Wolsey, L.: Cutting planes in integer and mixed integer programming. Discrete Applied Mathematics 123, 397–446 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Margot, F.: Testing Cut Generators for mixed-integer linear programming. Math. Prog. Comp. 1, 69–95 (2009)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. Wiley, Chichester (1988)MATHGoogle Scholar
  13. 13.
    Padberg, M., Rinaldi, G.: A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Reviews 33, 60–100 (1991)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Edoardo Amaldi
    • 1
  • Stefano Coniglio
    • 1
  • Stefano Gualandi
    • 1
  1. 1.Dipartimento di Elettronica e InformazionePolitecnico di MilanoItaly

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