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Improved Approximation Algorithms for Tverberg Partitions

Authors Sariel Har-Peled , Timothy Zhou



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Sariel Har-Peled
  • Department of Computer Science, University of Illinois, Urbana, IL, USA
Timothy Zhou
  • Department of Computer Science, University of Illinois, Urbana, IL, USA

Acknowledgements

The authors thank Timothy Chan, Wolfgang Mulzer, David Rolnick, and Pablo Soberon-Bravo for providing useful references.

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Sariel Har-Peled and Timothy Zhou. Improved Approximation Algorithms for Tverberg Partitions. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 51:1-51:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.51

Abstract

Tverberg’s theorem states that a set of n points in ℝ^d can be partitioned into ⌈n/(d+1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensional median, of the input point set. While randomized algorithms exist to find centerpoints with some failure probability, a partition for a Tverberg point provides a certificate of its correctness. Unfortunately, known algorithms for computing exact Tverberg points take n^{O(d²)} time. We provide several new approximation algorithms for this problem, which improve running time or approximation quality over previous work. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric spanners
  • vertex failures
  • robustness

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