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Monotone Diameter of Bisubmodular Polyhedra

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Abstract

Finding sharp bounds on the diameter of polyhedra is a fundamental problem in discrete mathematics and computational geometry. In particular, the monotone diameter and height play an important role in determining the number of iterations by operating the pivot rule of the simplex method for linear programming. In this study, for a d-dimensional polytope defined by at most \(3^{d} -1\) linear inequality induced by functions called bisubmodular, we prove that the diameter, monotone diameter, and height are coincide, and the tight upper bound is \({d}^2\).

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Notes

  1. We do not differentiate pivot rules for simplicity. It is clear that the height is the upper bound for any pivot rules.

  2. These conditions are to exclude the trivial case, where (9) apparently are satisfied since equalities \(\{(X_1,Y_1)\sqcup (X_2,Y_2),(X_1,Y_1)\sqcap (X_2,Y_2)\} = \{(X_1,Y_1), (X_2,Y_2) \}\).

  3. A special case, strict bisubmodular only, was briefly mentioned in [32].

  4. See also Theorem 8 and inequality (1).

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Acknowledgements

We are grateful to anonymous reviewers for their comments to improve the presentation of the present paper. We would like to thank RIMS, Kyoto University, for providing a chance of a presentation and a valuable discussion.

Funding

The first and third authors’ work were supported partially by JSPS KAKENHI Grant Number, 20K04973 and 20H05964, 20K04970, respectively.

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Authors and Affiliations

  1. Department of Mathematical Sciences, School of Science, Tokai University, 4-1-1, Kita-kaname, Hiratsuka, Kanagawa, 259-1292, Japan

    Yasuko Matsui

  2. Department of Advanced Sciences, Faculty of Science and Engineering, Hosei University, 3-7-2 Kajino-cho, Koganei, Tokyo, 184-8584, Japan

    Noriyoshi Sukegawa

  3. Department of Communication and Business, College of Media and Communication, Edogawa University, 474 Komaki, Nagareyama, Chiba, 270-0198, Japan

    Ping Zhan

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  1. Yasuko Matsui
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  2. Noriyoshi Sukegawa
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  3. Ping Zhan
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Correspondence to Ping Zhan.

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Matsui, Y., Sukegawa, N. & Zhan, P. Monotone Diameter of Bisubmodular Polyhedra. Oper. Res. Forum 4, 76 (2023). https://doi.org/10.1007/s43069-023-00260-1

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