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All Rational Polytopes Are Transportation Polytopes and All Polytopal Integer Sets Are Contingency Tables

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Integer Programming and Combinatorial Optimization (IPCO 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3064))

Abstract

We show that any rational polytope is polynomial-time representable as a “slim” r× c × 3 three-way line-sum transportation polytope. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure.

It provides polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bipartite biflow programs. It resolves several standing problems on 3-way transportation polytopes. It demonstrates that the range of values an entry can attain in any slim 3-way contingency table with specified 2-margins can contain arbitrary gaps, suggesting that disclosure of k-margins of d-tables for 2≤ k<d is confidential.

Our construction also provides a powerful tool in studying concrete questions about transportation polytopes and contingency tables; remarkably, it enables to automatically recover the famous “real-feasible integer-infeasible” 6× 4× 3 transportation polytope of M. Vlach, and to produce the first example of 2-margins for 6× 4× 3 contingency tables where the range of values a specified entry can attain has a gap.

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Author information

Authors and Affiliations

  1. University of California at Davis, Davis, CA, 95616, USA

    Jesus De Loera

  2. Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Shmuel Onn

Authors
  1. Jesus De Loera
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  2. Shmuel Onn
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Editor information

Editors and Affiliations

  1. Department of IEOR, Columbia University, 500 West 120th Street, 10027, New York, NY, USA

    Daniel Bienstock

  2. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    George Nemhauser

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© 2004 Springer-Verlag Berlin Heidelberg

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De Loera, J., Onn, S. (2004). All Rational Polytopes Are Transportation Polytopes and All Polytopal Integer Sets Are Contingency Tables. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_26

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  • DOI: https://doi.org/10.1007/978-3-540-25960-2_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22113-5

  • Online ISBN: 978-3-540-25960-2

  • eBook Packages: Springer Book Archive

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