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


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)


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
  • Geometric spanners
  • vertex failures
  • robustness


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  1. Peyman Afshani, Donald R. Sheehy, and Yannik Stein. Approximating the simplicial depth. CoRR, abs/1512.04856, 2015. URL:
  2. Pankaj K. Agarwal, Sariel Har-Peled, Subhash Suri, Hakan Yildiz, and Wuzhou Zhang. Convex hulls under uncertainty. Algorithmica, 79(2):340-367, 2017. URL:
  3. Pankaj K. Agarwal, Micha Sharir, and Emo Welzl. Algorithms for center and Tverberg points. ACM Trans. Algo., 5(1):5:1-5:20, 2008. URL:
  4. David Avis. The m-core properly contains the m-divisible points in space. Pattern Recognit. Lett., 14(9):703-705, 1993. URL:
  5. B. J. Birch. On 3N points in a plane. Mathematical Proceedings of the Cambridge Philosophical Society, 55(4):289–293, 1959. URL:
  6. T. M. Chan. Geometric applications of a randomized optimization technique. Disc. Comput. Geom., 22(4):547-567, 1999. URL:
  7. T. M. Chan. An optimal randomized algorithm for maximum Tukey depth. In J. Ian Munro, editor, Proc. 15th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 430-436. SIAM, 2004. URL:
  8. B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory, IT-31(4):509-517, 1985. URL:
  9. Aruni Choudhary and Wolfgang Mulzer. No-dimensional Tverberg theorems and algorithms. In Sergio Cabello and Danny Z. Chen, editors, Proc. 36th Int. Annu. Sympos. Comput. Geom. (SoCG), volume 164 of LIPIcs, pages 31:1-31:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
  10. K. L. Clarkson. Las Vegas algorithms for linear and integer programming. J. Assoc. Comput. Mach., 42:488-499, 1995. URL:
  11. Kenneth L. Clarkson, David Eppstein, Gary L. Miller, Carl Sturtivant, and Shang-Hua Teng. Approximating center points with iterative Radon points. Int. J. Comput. Geom. Appl., 6:357-377, 1996. URL:
  12. Ketan Dalal. Counting the onion. Random Struct. Alg., 24(2):155-165, 2004. URL:
  13. S. Har-Peled and B. Lidicky. Peeling the grid. SIAM J. Discrete Math., 27(2):650-655, 2013. URL:
  14. S. Har-Peled and M. Sharir. Relative (p,ε)-approximations in geometry. Disc. Comput. Geom., 45(3):462-496, 2011. URL:
  15. Sariel Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, 2011. Google Scholar
  16. Sariel Har-Peled and Mitchell Jones. Journey to the center of the point set. In Gill Barequet and Yusu Wang, editors, Proc. 35th Int. Annu. Sympos. Comput. Geom. (SoCG), volume 129 of LIPIcs, pages 41:1-41:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL:
  17. Sariel Har-Peled and Timothy Zhou. Improved approximation algorithms for Tverberg partitions. CoRR, abs/2007.08717, 2020. URL:
  18. D. Haussler and E. Welzl. ε-nets and simplex range queries. Disc. Comput. Geom., 2:127-151, 1987. URL:
  19. Shreesh Jadhav and Asish Mukhopadhyay. Computing a centerpoint of a finite planar set of points in linear time. Disc. Comput. Geom., 12:291-312, 1994. URL:
  20. Yin Tat Lee and Aaron Sidford. Efficient inverse maintenance and faster algorithms for linear programming. In Venkatesan Guruswami, editor, Proc. 56th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 230-249. IEEE Computer Society, 2015. URL:
  21. J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16(4/5):498-516, 1996. URL:
  22. Frédéric Meunier, Wolfgang Mulzer, Pauline Sarrabezolles, and Yannik Stein. The rainbow at the end of the line - A PPAD formulation of the colorful carathéodory theorem with applications. In Philip N. Klein, editor, Proc. 28th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 1342-1351. SIAM, 2017. URL:
  23. Gary L. Miller and Donald R. Sheehy. Approximate centerpoints with proofs. Comput. Geom., 43(8):647-654, 2010. URL:
  24. Wolfgang Mulzer and Daniel Werner. Approximating Tverberg points in linear time for any fixed dimension. Disc. Comput. Geom., 50(2):520-535, 2013. URL:
  25. John R. Reay. Several generalizations of Tverberg’s theorem. Israel Journal of Mathematics, 34(3):238-244, 1979. URL:
  26. David Rolnick and Pablo Soberón. Algorithms for Tverberg’s theorem via centerpoint theorems. CoRR, abs/1601.03083, 2016. URL:
  27. R. Seidel. Small-dimensional linear programming and convex hulls made easy. Disc. Comput. Geom., 6:423-434, 1991. URL:
  28. Helge Tverberg and Siniša Vrećica. On generalizations of Radon’s theorem and the ham sandwich theorem. Euro. J. Combin., 14(3):259-264, 1993. URL:
  29. Jan van den Brand, Yin Tat Lee, Aaron Sidford, and Zhao Song. Solving tall dense linear programs in nearly linear time. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proc. 52nd Annu. ACM Sympos. Theory Comput. (STOC), pages 775-788. ACM, 2020. URL: