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)}).

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doi.org/10.4230/LIPIcs.SoCG.2022.18
Towards a Quantitative Theory of Integer Programming
Annual Symposium on Computational Geometry
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bonnet, G.F.Y, Dadush, D.N, Grupel, U, Huiberts, S, & Livshyts, G. (2022). Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes. In Leibniz International Proceedings in Informatics (pp. 18:1–18:15). doi:10.4230/LIPIcs.SoCG.2022.18