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Bipartite Designs

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Integer Programming and Combinatorial Optimization (IPCO 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1412))

Abstract

We investigate the solution set of the following matrix equation: A T B = J +diag(d), where A and B are n × n {0, 1} matrices, J is the matrix with all entries equal one and d is a full support vector. We prove that in some special cases (such as: both A d −1 and B d −1 have full supports, where d −1 = (d −11 , ..., d −1n )T; both A and B have constant column sums; d −1 · 1 ≠ −1, and A has constant row sum etc.) these solutions have strong structural properties. We show how the results relate to design theory, and then apply the results to derive sharper characterizations of (α,ω)-graphs. We also deduce consequences for “minimal” polyhedra with {0, 1} vertices having non-{0, 1} constraints, and “minimal” polyhedra with {0, 1} constraints having non-{0, 1} vertices.

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Author information

Authors and Affiliations

  1. Yerevan State University, Yerevan-49, Armenia

    Grigor Gasparyan

Authors
  1. Grigor Gasparyan
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Editors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6020 Annapolis, Houston, TX, 77005, USA

    Robert E. Bixby

  2. PROS Strategic Solutions, 3223 Smith Street, Houston, TX, 77006, USA

    E. Andrew Boyd

  3. Department of Industrial, Engineering, Texas A&M University, 1000 Country Place Dr. Apt. 69, Houston, TX, 77079, USA

    Roger Z. Ríos-Mercado

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© 1998 Springer-Verlag Berlin Heidelberg

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Gasparyan, G. (1998). Bipartite Designs. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization. IPCO 1998. Lecture Notes in Computer Science, vol 1412. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-69346-7_3

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  • DOI: https://doi.org/10.1007/3-540-69346-7_3

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64590-0

  • Online ISBN: 978-3-540-69346-8

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