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Proving total dual integrality with cross-free families—A general framework

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Abstract

We present a theorem stating that certain classes of linear programming problems have integer optimal (primal and dual) solutions. The theorem includes as special cases earlier results of Johnson, Edmonds and Giles, Frank, Hoffman and Schwartz, Gröflin and Hoffman, and Lawler and Martel. The proof method consists of deriving total dual integrality for the corresponding system of linear inequalities from the total unimodularity of certain ‘cross-free’ subsystems. The scheme presented here differs from the one proposed earlier by Grishuhin in that Grishuhin requires the total unimodularity of cross-free subsystems in the axioms, whereas here this follows from easier verifiable axioms.

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Authors and Affiliations

  1. Department of Econometrics, Tilburg University, Postbus 90153, 5000 LE, Tilburg

    A. Schrijver

  2. Department of Operations Research, Mathematical Centre, Kruislaan 413, 1098 SJ, Amsterdam, Holland

    A. Schrijver

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  1. A. Schrijver
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Schrijver, A. Proving total dual integrality with cross-free families—A general framework. Mathematical Programming 29, 15–27 (1984). https://doi.org/10.1007/BF02591726

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  • DOI: https://doi.org/10.1007/BF02591726

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