Improved Approximation Algorithms for Tverberg Partitions

07/17/2020

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by   Sariel Har-Peled, et al.

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Tverberg's theorem states that a set of n points in ^d can be partitioned into n/(d+1) sets with a common intersection. A point in this intersection (aka Tverberg point) is a centerpoint of the input point set, and the Tverberg partition provides a compact proof of this, which is algorithmically useful. Unfortunately, computing a Tverberg point exactly requires n^O(d^2) time. We provide several new approximation algorithms for this problem, which improve either the running time or quality of approximation, or both. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.