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Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

Authors Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, Galyna Livshyts

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Author Details

Gilles Bonnet
  • University of Groningen, The Netherlands
Daniel Dadush
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Uri Grupel
  • Universität Innsbruck, Austria
Sophie Huiberts
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Galyna Livshyts
  • Georgia Institute of Technology, Atlanta, GA, USA

Funding

  • Bonnet, Gilles: Funded by the DFG Priority Program (SPP) 2265 https://spp2265.wias-berlin.de/index.php, project P23.
  • Dadush, Daniel: Supported by the ERC Starting grant QIP-805241.

Acknowledgements

This work was done in part while the authors were participating in the Probability, Geometry and Computation in High Dimensions semester at the Simons Institute for the Theory of Computing and in the Interplay between High-Dimensional Geometry and Probability trimester at the Hausdorff Institute for Mathematics.

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Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, and Galyna Livshyts. Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.18

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 n-dimensional polytope P defined by the intersection of m i.i.d. half-spaces whose normals are chosen uniformly from the sphere, we show that diam(P) is Ω(n m^{1/(n-1)}) and O(n² m^{1/(n-1)} + 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/(n-1)}) 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 worst-case 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) ≥ (n-1)(diam(P^∘)-2). We then prove that the shortest path between any "nearly" antipodal pair vertices of P^∘ has length Ω(m^{1/(n-1)}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Random Polytopes
  • Combinatorial Diameter
  • Hirsch Conjecture

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