Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes
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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(n^2 m^1/n-1 + n^5 4^n) 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(n^2 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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