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On the Maximal Number of Columns of a \(\varDelta \)-modular Matrix

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Integer Programming and Combinatorial Optimization (IPCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13265))

Abstract

We study the maximal number of pairwise distinct columns in a \(\varDelta \)-modular integer matrix with m rows. Recent results by Lee et al. provide an asymptotically tight upper bound of \(\mathcal {O}(m^2)\) for fixed \(\varDelta \). We complement this and obtain an upper bound of the form \(\mathcal {O}(\varDelta )\) for fixed m, and with the implied constant depending polynomially on m.

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Notes

  1. 1.

    The authors of [4, 7] use the term \(\varDelta \)-modular for what we call \(\varDelta \)-submodular.

  2. 2.

    Lee et al. [7, p. 23] remark that their techniques provide \({{\,\mathrm{h}\,}}(\varDelta ,m) \le \mathcal {O}(m^2 \cdot \varDelta ^{1.95})\).

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Authors and Affiliations

  1. BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany

    Gennadiy Averkov & Matthias Schymura

Authors
  1. Gennadiy Averkov
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  2. Matthias Schymura
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Correspondence to Matthias Schymura .

Editor information

Editors and Affiliations

  1. Delft University of Technology, Delft, The Netherlands

    Karen Aardal

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Laura Sanità

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Averkov, G., Schymura, M. (2022). On the Maximal Number of Columns of a \(\varDelta \)-modular Matrix. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_3

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  • DOI: https://doi.org/10.1007/978-3-031-06901-7_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-06900-0

  • Online ISBN: 978-3-031-06901-7

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