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An Asymptotically Improved Upper Bound on the Diameter of Polyhedra

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Abstract

Kalai and Kleitman proved in 1992 that the maximum possible diameter of a d-dimensional polyhedron with n facets is at most \(n^{2+ \log _2 d}\). In 2014, Todd improved the Kalai–Kleitman bound to \((n-d)^{\log _2 d}\). Todd’s bound is tight for \(d \le 2\), and has been improved for \(d \ge 3\) in subsequent studies. The current best upper bound is, however, still in the form of \((n-d)^{\log _2 \mathrm{O}(d )}\). In this paper, we show an asymptotically improved upper bound of \((n-d)^{\log _2 \mathrm{O}(d/{\log }\, d )}\).

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Acknowledgements

The author is grateful to Kazuo Murota and Yoshio Okamoto for stimulating this study. The author is also grateful to Antoine Deza, Yuya Higashikawa, Lionel Pournin, and two anonymous referees for their helpful comments. This research is supported in part by Grant-in-Aid for Science Research (A) 26242027.

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  1. Department of Information and Computer Technology, Tokyo University of Science, 6-3-1, Niijuku, Katsushika-ku, Tokyo, 125-8585, Japan

    Noriyoshi Sukegawa

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  1. Noriyoshi Sukegawa
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Correspondence to Noriyoshi Sukegawa.

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Sukegawa, N. An Asymptotically Improved Upper Bound on the Diameter of Polyhedra. Discrete Comput Geom 62, 690–699 (2019). https://doi.org/10.1007/s00454-018-0016-y

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