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Simpler exchange axioms for M-concave functions on generalized polymatroids

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Abstract

\({\mathrm{M}}^\natural \)-concave functions form a class of discrete concave functions in discrete convex analysis, and are defined by a certain exchange axiom. We show in this paper that \({\mathrm{M}}^\natural \)-concave functions can be characterized by a combination of two simpler exchange properties. It is also shown that for a function defined on an integral polymatroid, a much simpler exchange axiom characterizes \({\mathrm{M}}^\natural \)-concavity. These results have some significant implications in discrete convex analysis.

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Notes

  1. The definition of an integral polymatroid will be given in Sect. 2.

  2. \(\mathbb {B}\) stands for Binary, referring to functions on \(\{0,1\}^n\).

  3. In [3] the concept of well-layered map is defined for more general set functions for which the effective domain can be a proper subset of \(2^N\). We here restrict our attention to the case of \(\mathrm{dom_{\mathbb {B}}\,}f = 2^N\) for simplicity of description.

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Acknowledgements

The authors thank Yu Yokoi and the anonymous referees for their valuable comments on the manuscript.

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Authors and Affiliations

  1. School of Business Administration, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Kazuo Murota

  2. Department of Industrial Engineering and Economics, Tokyo Institute of Technology, Tokyo, 152-8550, Japan

    Akiyoshi Shioura

Authors
  1. Kazuo Murota
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  2. Akiyoshi Shioura
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Corresponding author

Correspondence to Akiyoshi Shioura.

Additional information

This work was supported by The Mitsubishi Foundation, CREST, JST, Grant number JPMJCR14D2, Japan, and JSPS KAKENHI Grant numbers 26280004, 15K00030, 15H00848.

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Murota, K., Shioura, A. Simpler exchange axioms for M-concave functions on generalized polymatroids. Japan J. Indust. Appl. Math. 35, 235–259 (2018). https://doi.org/10.1007/s13160-017-0285-5

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  • DOI: https://doi.org/10.1007/s13160-017-0285-5

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