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Algorithms for Tolerated Tverberg Partitions

  • Wolfgang Mulzer
  • Yannik Stein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)

Abstract

Let P be a d-dimensional n-point set. A partition \(\mathcal{T}\) of P is called a Tverberg partition if the convex hulls of all sets in \(\mathcal{T}\) intersect in at least one point. We say \(\mathcal{T}\) is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with ⌈n / (d + 1)(t + 1) ⌉ sets. However, so far no nontrivial algorithms for computing or approximating such partitions have been presented.

For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the (untolerated) Tverberg problem. Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerated.

Keywords

Approximation Algorithm Convex Hull London Mathematical Society Lift Step Mial Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wolfgang Mulzer
    • 1
  • Yannik Stein
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany

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