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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chan, T.M.: An optimal randomized algorithm for maximum Tukey depth. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 430–436 (2004)Google Scholar
  2. 2.
    Clarkson, K.L., Eppstein, D., Miller, G.L., Sturtivant, C., Hua Teng, S.: Approximating center points with iterative Radon points. International Journal of Computational Geometry & Applications 6, 357–377 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)Google Scholar
  4. 4.
    García Colín, N.: Applying Tverberg Type Theorems to Geometric Problems. PhD thesis, University College London (2007)Google Scholar
  5. 5.
    Larman, D.: On sets projectively equivalent to the vertices of a convex polytope. Bulletin of the London Mathematical Society 4(1), 6–12 (1972)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Matoušek, J.: Lectures on Discrete Geometry, 1st edn. Springer (2002)Google Scholar
  7. 7.
    Miller, G.L., Sheehy, D.R.: Approximate centerpoints with proofs. Computational Geometry 43, 647–654 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Montejano, L., Oliveros, D.: Tolerance in Helly-type theorems. Discrete & Computational Geometry 45, 348–357 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mulzer, W., Werner, D.: Approximating Tverberg points in linear time for any fixed dimension. In: Proceedings of the 28th Annual Symposium on Computational Geometry, pp. 303–310 (2012)Google Scholar
  10. 10.
    Rado, R.: A theorem on general measure. Journal of the London Mathematical Society 1, 291–300 (1946)CrossRefGoogle Scholar
  11. 11.
    Sarkaria, K.: Tverberg’s theorem via number fields. Israel Journal of Mathematics 79, 317–320 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Soberón, P., Strausz, R.: A generalisation of Tverberg’s theorem. Discrete & Computational Geometry 47, 455–460 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Teng, S.-H.: Points, spheres, and separators: a unified geometric approach to graph partitioning. PhD thesis, Carnegie Mellon University Pittsburgh (1992)Google Scholar
  14. 14.
    Tverberg, H.: A generalization of Radon’s theorem. Journal of the London Mathematical Society 41, 123–128 (1966)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

Personalised recommendations