Transitive packing

  • Rudolf Müller
  • Andreas S. Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1084)

Abstract

This paper is intended to give a concise understanding of the facial structure of previously separately investigated polyhedra. We introduce the notion of transitive packing and the transitive packing polytope and give cutting plane proofs for huge classes of valid inequalities of this polytope. We introduce generalized cycle, generalized clique, generalized antihole, generalized antiweb, generalized web, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases but also give many new inequalities for several others. For some of the classes we also prove a nontrivial lower bound for their Chvátal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices.

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References

  1. [BGM85]
    F. Barahona, M. Grötschel, and A. R. Mahjoub. Facets of the bipartite subgraph polytope. Mathematics of Operations Research, 10:340–358, 1985.Google Scholar
  2. [BM86]
    F. Barahona and A. R. Mahjoub. On the cut polytope. Mathematical Programming, 36:157–173, 1986.Google Scholar
  3. [BN89a]
    E. Balas and S. M. Ng. On the set covering polytope: I. AH the facets with coefficients in {0,1, 2}. Mathematical Programming, 43:57–69, 1989.CrossRefGoogle Scholar
  4. [BN89b]
    E. Balas and S. M. Ng. On the set covering polytope: II. Lifting the facets with coefficients in {0,1,2}. Mathematical Programming, 45:1–20, 1989.CrossRefGoogle Scholar
  5. [BP76]
    E. Balas and M. W. Padberg. Set partitioning: A survey. SIAM Review, 18:710–760, 1976.CrossRefGoogle Scholar
  6. [CC92]
    M. Conforti and G. Cornuéjols. Balanced 0, ±1 matrices, bicoloring and total dual integrality. Preprint, Carnegie Mellon University, Pittsburgh, USA, 1992.Google Scholar
  7. [CCH89]
    V. Chvátal, W. Cook, and M. Hartmann. On cutting-plane proofs in combinatorial optimization. Linear Algebra and its Applications, 114/115:455–499, 1989.CrossRefGoogle Scholar
  8. [CF95]
    A. Caprara and M. Fischetti. 0, 1/2-Chvátal-Gomory cuts. Technical Report, DEIS, University of Bologna, Bologna, Italy, 1993, revised 1995.Google Scholar
  9. [Chv73]
    V. Chvátal. Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics, 4:305–337, 1973.CrossRefGoogle Scholar
  10. [CR93]
    S. Chopra and M. R. Rao. The partition problem. Mathematical Programming, 59:87–115, 1993.CrossRefGoogle Scholar
  11. [CS89]
    G. Cornuéjols and A. Sassano. On the 0,1 facets of the set covering polytope. Mathematical Programming, 43:45–55, 1989.CrossRefGoogle Scholar
  12. [Edm62]
    J. Edmonds. Covers and packings in a family of sets. Bulletin of the American Mathematical Society, 68:494–499, 1962.Google Scholar
  13. [EJR87]
    R. Euler, M. Jünger, and G. Reinelt. Generalizations of cliques, odd cycles and anticycles and their relation to independence system polyhedra. Mathematics of Operations Research, 12:451–462, 1987.Google Scholar
  14. [GJR85a]
    M. Grötschel, M. Jünger, and G. Reinelt. Acyclic subdigraphs and linear orderings: Polytopes, facets, and cutting plane algorithms. In I. Rival, editor, Graphs and Order, pages 217–266. D. Reidel Publishing Company, Dordrecht, 1985.Google Scholar
  15. [GJR85b]
    M. Grötschel, M. Jünger, and G. Reinelt. On the acyclic subgraph polytope. Mathematical Programming, 33:28–42, 1985.CrossRefGoogle Scholar
  16. [GLS81]
    M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169–197, 1981. (Corrigendum: 4 (1984), 291–295).Google Scholar
  17. [GLS88]
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin, 1988.Google Scholar
  18. [GW89]
    M. Grötschel and Y. Wakabayashi. A cutting plane algorithm for a clustering problem. Mathematical Progamming, 45:59–96, 1989.CrossRefGoogle Scholar
  19. [GW90]
    M. Grötschel and Y. Wakabayashi. Facets of the clique partitioning polytope. Mathematical Programming. 47:367–388, 1990.CrossRefGoogle Scholar
  20. [JM93]
    M. Jünger and P. Mutzel. Solving the maximum weight planar subgraph. In G. Rinaldi and L. A, Wolsey, editors, Integer Programming and Combinatorial Optimization, pages 479–492, 1993. Proceedings of the 3rd International IPCO Conference.Google Scholar
  21. [Jün85]
    M. Jünger. Polyhedral Combinatorics and the Acyclic Subdigraph Problem, volume 7 of Research and Expositions in Mathematics. Heldermann Verlag Berlin, 1985.Google Scholar
  22. [KL89]
    B. Korte and L. Lovász. Polyhedral results for antimatroids. In G. S. Bloom, R. L. Graham, and J. Malkevitch, editors, Combinatorial Mathematics, pages 283–295. Academy of Sciences, New York, 1989. Proceedings of the Third International Conference.Google Scholar
  23. [Lau89]
    M. Laureat. A generalization of antiwebs to independence systems and their canonical facets. Mathematical Programming, 45:97–108, 1989.CrossRefGoogle Scholar
  24. [MS95]
    R. Müller and A. S. Schulz. The interval order polytope of a digraph. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, number 920 in Lecture Notes in Computer Science, pages 50–64. Springer, Berlin, 1995. Proceedings of the 4th International IPCO Conference.Google Scholar
  25. [MS96]
    R. Müller and A. S. Schulz. Transitive packing. Preprint, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1996.Google Scholar
  26. [Mül93]
    R. Müller. On the transitive acyclic subdigraph polytope. In G. Rinaldi and L. A. Wolsey, editors, Integer Programming and Combinatorial Optimization, pages 463–477, 1993. Proceedings of the 3rd International IPCO Conference.Google Scholar
  27. [NS89]
    P. Nobili and A. Sassano. Facets and lifting procedures for the set covering polytope. Mathematical Programming, 45:111–137, 1989.CrossRefGoogle Scholar
  28. [NS95]
    P. Nobili and A. Sassano. (0, ±1) ideal matrices. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, number 920 in Lecture Notes in Computer Science, pages 344–359. Springer, Berlin, 1995. Proceedings of the 4th International IPCO Conference.Google Scholar
  29. [NT74]
    G. L. Nemhauser and L. E. Trotter Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6:48–61, 1974.CrossRefGoogle Scholar
  30. [NW88]
    G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley & Sons, New York, 1988.Google Scholar
  31. [ORS95]
    M. Oosten, J. H. G. C. Rutten, and F. C. R. Spieksma. The clique partitioning polytope: Facets. Department of Mathematics, University of Limburg, Maastricht, The Netherlands, 1995.Google Scholar
  32. [Pad73]
    M. W. Padberg. On the facial structure of set packing polyhedra. Mathematical Programming, 5:199–215, 1973.CrossRefGoogle Scholar
  33. [Pad79]
    M. W. Padberg. Covering, packing and knapsack problems. Annals of Discrete Mathematics, 4:265–287, 1979.MathSciNetGoogle Scholar
  34. [SasS9]
    A. Sassano. On the facial structure of the set covering polytope. Mathematical Programming, 44:181–202, 1989.CrossRefGoogle Scholar
  35. [SB]
    D. F. Shallcross and R. G. Bland. On the polyhedral structure of relatively transitive subgraphs. Technical report, Cornell University, Ithaca, NY.Google Scholar
  36. [Sch80]
    A. Schrijver. On cutting planes. In M. Deza and I. G. Rosenberg, editors, Combinatorics '79, Part II, volume 9 of 'Annals of Discrete Mathematics, pages 291–296. North-Holland, Amsterdam, 1980.Google Scholar
  37. [Sch86]
    A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, Chichester, 1986.Google Scholar
  38. [Sch95]
    A. S. Schulz. Polytopes and Scheduling. PhD thesis, Technical University of Berlin, Berlin, Germany, 1995.Google Scholar
  39. [Tro75]
    L. E. Trotter Jr. A class of facet producing graphs for vertex packing polyhedra. Discrete Mathematics, 12:373–388, 1975.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Rudolf Müller
    • 1
  • Andreas S. Schulz
    • 2
  1. 1.Institut für WirtschaftsinformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fachbereich MathematikTechnische Universität BerlinBerlinGermany

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