Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

Abstract : Convex bodies play a fundamental role in geometric computation, and approximating such bodies is often a key ingredient in the design of efficient algorithms. We consider the question of how to succinctly approximate a multidimensional convex body by a polytope. We are given a convex body K of unit diameter in Euclidean d-dimensional space (where d is a constant) along with an error parameter ε > 0. The objective is to determine a polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. In the mid-1970's, a result by Dudley showed that O(1/ε (d−1)/2) facets suffice, and Bronshteyn and Ivanov presented a similar bound on the number of vertices. While both results match known worst-case lower bounds, obtaining a similar upper bound on the total combinatorial complexity has been open for over 40 years. Recently, we made a first step forward towards this objective, obtaining a suboptimal bound. In this paper, we settle this problem with an asymptotically optimal bound of O(1/ε (d−1)/2). Our result is based on a new relationship between ε-width caps of a convex body and its polar. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are "essentially different." We achieve our result by combining this with a variant of the witness-collector method and a novel variable-width layered construction.
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https://hal.archives-ouvertes.fr/hal-02440482
Contributor : Guilherme D. da Fonseca <>
Submitted on : Wednesday, January 15, 2020 - 11:07:13 AM
Last modification on : Thursday, January 23, 2020 - 1:40:35 AM

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Rahul Arya, Sunil Arya, Guilherme da Fonseca, David Mount. Optimal Bound on the Combinatorial Complexity of Approximating Polytopes. SODA 2020, Jan 2020, Salt Lake City, United States. pp.786-805, ⟨10.1137/1.9781611975994.48⟩. ⟨hal-02440482⟩

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