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DOI: 10.4230/LIPIcs.ESA.2021.51

Improved Approximation Algorithms for Tverberg Partitions

Authors: Sariel Har-Peled and Timothy Zhou

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

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.

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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)

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@InProceedings{harpeled_et_al:LIPIcs.ESA.2021.51,
  author =	{Har-Peled, Sariel and Zhou, Timothy},
  title =	{{Improved Approximation Algorithms for Tverberg Partitions}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{51:1--51:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.51},
  URN =		{urn:nbn:de:0030-drops-146323},
  doi =		{10.4230/LIPIcs.ESA.2021.51},
  annote =	{Keywords: Geometric spanners, vertex failures, robustness}
}

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Improved Approximation Algorithms for Tverberg Partitions

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