Abstract
The combinatorial diameter diam(P) of a polytope P is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an ndimensional polytope P defined by the intersection of m i.i.d. halfspaces whose normals are chosen uniformly from the sphere, we show that diam(P) is Ω(n m^{1/(n1)}) and O(n² m^{1/(n1)} + n⁵ 4ⁿ) with high probability when m ≥ 2^{Ω(n)}.
For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when m is large, where we rely on the Θ(n² m^{1/(n1)}) bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these "shadows paths" together over a suitable net using worstcase diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope P^∘, corresponding to a random convex hull, by showing the relation diam(P) ≥ (n1)(diam(P^∘)2). We then prove that the shortest path between any "nearly" antipodal pair vertices of P^∘ has length Ω(m^{1/(n1)}).
BibTeX  Entry
@InProceedings{bonnet_et_al:LIPIcs.SoCG.2022.18,
author = {Bonnet, Gilles and Dadush, Daniel and Grupel, Uri and Huiberts, Sophie and Livshyts, Galyna},
title = {{Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {18:118:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772273},
ISSN = {18688969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16026},
URN = {urn:nbn:de:0030drops160269},
doi = {10.4230/LIPIcs.SoCG.2022.18},
annote = {Keywords: Random Polytopes, Combinatorial Diameter, Hirsch Conjecture}
}
Keywords: 

Random Polytopes, Combinatorial Diameter, Hirsch Conjecture 
Collection: 

38th International Symposium on Computational Geometry (SoCG 2022) 
Issue Date: 

2022 
Date of publication: 

01.06.2022 