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Integral Polyhedra Associated with Certain Submodular Functions Defined on 012-Vectors

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Integer Programming and Combinatorial Optimization (IPCO 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1610))

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Abstract

A new class of polyhedra, named greedy-type polyhedra, is introduced. This class contains polyhedra associated with submodular set functions. Greedy-type polyhedra are associated with submodular functions defined on 012-vectors and have 012-vectors as normal vectors of their facets. The face structure of greedy-type polyhedra is described with maximal chains of a certain partial order defined on 012-vectors. Integrality of polyhedra associated with integral greedy-type functions is shown through total dual integrality of the systems of inequalities defining polyhedra. Then a dual algorithm maximizing linear functions over these polyhedra is proposed. It is shown that feasible outputs of certain bipartite networks with gain make greedy-type polyhedra. A separation theorem for greedy-type functions is also proved.

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Authors and Affiliations

  1. Graduate Division of International and Interdisciplinary Studies, University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8902, Japan

    Kenji Kashiwabara, Masataka Nakamura & Takashi Takabatake

Authors
  1. Kenji Kashiwabara
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  2. Masataka Nakamura
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  3. Takashi Takabatake
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Editor information

Editors and Affiliations

  1. GSIA, Carnegie Mellon University, Schenley Park, Pittsburgh, PA, 15213, USA

    Gérard Cornuéjols

  2. Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Rainer E. Burkard  & Gerhard J. Woeginger  & 

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© 1999 Springer-Verlag Berlin Heidelberg

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Kashiwabara, K., Nakamura, M., Takabatake, T. (1999). Integral Polyhedra Associated with Certain Submodular Functions Defined on 012-Vectors. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds) Integer Programming and Combinatorial Optimization. IPCO 1999. Lecture Notes in Computer Science, vol 1610. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48777-8_22

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  • DOI: https://doi.org/10.1007/3-540-48777-8_22

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66019-4

  • Online ISBN: 978-3-540-48777-7

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