# Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes

@inproceedings{Bonnet2021AsymptoticBO, title={Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes}, author={Gilles Bonnet and Daniel Dadush and Uri Grupel and Sophie Huiberts and Galyna V. Livshyts}, booktitle={International Symposium on Computational Geometry}, year={2021} }

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…

## 2 Citations

### Small Shadows of Lattice Polytopes

- Mathematics
- 2022

. The diameter of the graph of a d -dimensional lattice polytope P ⊆ [0 ,k ] n is known to be at most dk due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone…

### Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method

- Computer ScienceArXiv
- 2022

This work proves that the smoothed complexity of the simplex method is O ( σ − 3 / 2 d 13 / 4 log 7 / 4 n ), improving the dependence on 1 /σ compared to the previous bound of O (σ − 2 d 2 √ log n ).

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