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… 

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References

SHOWING 1-10 OF 40 REFERENCES

A Spectral Approach to Polytope Diameter

Upper bounds on the graph diameters of polytopes are proved in two settings: a smoothed analysis bound and a spectral gap bounds arising from the log-concavity of the volume of a simple polytope in terms of its slack variables.

The Diameter of the Fractional Matching Polytope and Its Hardness Implications

  • Laura Sanità
  • Mathematics
    2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2018
It is shown that the problem of computing the diameter is strongly NP-hard even for apolytope with a very simple structure: namely, the fractional matching polytope, and that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytopes is an APX-hard problem.

On the diameter of convex polytopes

A quasi-polynomial bound for the diameter of graphs of polyhedra

The diameter of the graph of a d-dimensional polyhedron with n facets is at most nlog d+2 Let P be a convex polyhedron. The graph of P denoted by G(P ) is an abstract graph whose vertices are the

The Covering Radius of Randomly Distributed Points on a Manifold

We derive fundamental asymptotic results for the expected covering radius ρ(XN) for N points that are randomly and independently distributed with respect to surface measure on a sphere as well as on

Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method

  • R. Vershynin
  • Mathematics
    2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)
  • 2006
It is shown that the length of walk is actually polylogarithmic in the number of constraints n, and improved Spielman-Teng's bound on the walk O*(n86d 55sigma-30) to O(max(d5log2n, d9 log4d, d 3sigma -4)).

Hirsch polytopes with exponentially long combinatorial segments

A formulation of combinatorial segments is proposed which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes.

Deterministic and randomized polynomial-time approximation of radii

This paper is concerned with convex bodies in n-dimensional l p spaces, where each body is accessible only by a weak separation or optimization oracle. It studies the asymptotic relative accuracy, as

An upper bound for the diameter of a polytope

On the shape of the convex hull of random points

SummaryDenote by En the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that En has exactly n vertices. It is proved here