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Lattice polyhedra and submodular flows

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Abstract

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Satoru Fujishige

  2. TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Britta Peis

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  1. Satoru Fujishige
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  2. Britta Peis
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Correspondence to Britta Peis.

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Fujishige, S., Peis, B. Lattice polyhedra and submodular flows. Japan J. Indust. Appl. Math. 29, 441–451 (2012). https://doi.org/10.1007/s13160-012-0084-y

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  • DOI: https://doi.org/10.1007/s13160-012-0084-y

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