Transitive packing
 Rudolf Müller,
 Andreas S. Schulz
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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.
 F. Barahona, M. Grötschel, and A. R. Mahjoub. Facets of the bipartite subgraph polytope. Mathematics of Operations Research, 10:340–358, 1985.
 F. Barahona and A. R. Mahjoub. On the cut polytope. Mathematical Programming, 36:157–173, 1986.
 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. CrossRef
 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. CrossRef
 E. Balas and M. W. Padberg. Set partitioning: A survey. SIAM Review, 18:710–760, 1976. CrossRef
 M. Conforti and G. Cornuéjols. Balanced 0, ±1 matrices, bicoloring and total dual integrality. Preprint, Carnegie Mellon University, Pittsburgh, USA, 1992.
 V. Chvátal, W. Cook, and M. Hartmann. On cuttingplane proofs in combinatorial optimization. Linear Algebra and its Applications, 114/115:455–499, 1989. CrossRef
 A. Caprara and M. Fischetti. 0, 1/2ChvátalGomory cuts. Technical Report, DEIS, University of Bologna, Bologna, Italy, 1993, revised 1995.
 V. Chvátal. Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics, 4:305–337, 1973. CrossRef
 S. Chopra and M. R. Rao. The partition problem. Mathematical Programming, 59:87–115, 1993. CrossRef
 G. Cornuéjols and A. Sassano. On the 0,1 facets of the set covering polytope. Mathematical Programming, 43:45–55, 1989. CrossRef
 J. Edmonds. Covers and packings in a family of sets. Bulletin of the American Mathematical Society, 68:494–499, 1962.
 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.
 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.
 M. Grötschel, M. Jünger, and G. Reinelt. On the acyclic subgraph polytope. Mathematical Programming, 33:28–42, 1985. CrossRef
 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).
 M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, Berlin, 1988.
 M. Grötschel and Y. Wakabayashi. A cutting plane algorithm for a clustering problem. Mathematical Progamming, 45:59–96, 1989. CrossRef
 M. Grötschel and Y. Wakabayashi. Facets of the clique partitioning polytope. Mathematical Programming. 47:367–388, 1990. CrossRef
 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.
 M. Jünger. Polyhedral Combinatorics and the Acyclic Subdigraph Problem, volume 7 of Research and Expositions in Mathematics. Heldermann Verlag Berlin, 1985.
 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.
 M. Laureat. A generalization of antiwebs to independence systems and their canonical facets. Mathematical Programming, 45:97–108, 1989. CrossRef
 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.
 R. Müller and A. S. Schulz. Transitive packing. Preprint, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1996.
 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.
 P. Nobili and A. Sassano. Facets and lifting procedures for the set covering polytope. Mathematical Programming, 45:111–137, 1989. CrossRef
 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.
 G. L. Nemhauser and L. E. Trotter Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6:48–61, 1974. CrossRef
 G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley & Sons, New York, 1988.
 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.
 M. W. Padberg. On the facial structure of set packing polyhedra. Mathematical Programming, 5:199–215, 1973. CrossRef
 M. W. Padberg. Covering, packing and knapsack problems. Annals of Discrete Mathematics, 4:265–287, 1979.
 A. Sassano. On the facial structure of the set covering polytope. Mathematical Programming, 44:181–202, 1989. CrossRef
 D. F. Shallcross and R. G. Bland. On the polyhedral structure of relatively transitive subgraphs. Technical report, Cornell University, Ithaca, NY.
 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. NorthHolland, Amsterdam, 1980.
 A. Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, Chichester, 1986.
 A. S. Schulz. Polytopes and Scheduling. PhD thesis, Technical University of Berlin, Berlin, Germany, 1995.
 L. E. Trotter Jr. A class of facet producing graphs for vertex packing polyhedra. Discrete Mathematics, 12:373–388, 1975. CrossRef
 Title
 Transitive packing
 Book Title
 Integer Programming and Combinatorial Optimization
 Book Subtitle
 5th International IPCO Conference Vancouver, British Columbia, Canada, June 3–5, 1996 Proceedings
 Pages
 pp 430444
 Copyright
 1996
 DOI
 10.1007/3540613102_32
 Print ISBN
 9783540613107
 Online ISBN
 9783540684534
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1084
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
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 Editors
 Authors

 Rudolf Müller ^{(1)}
 Andreas S. Schulz ^{(2)}
 Author Affiliations

 1. Institut für Wirtschaftsinformatik, HumboldtUniversität zu Berlin, D10178, Berlin, Germany
 2. Fachbereich Mathematik, Technische Universität Berlin, D10623, Berlin, Germany
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