No-Dimensional Tverberg Theorems and Algorithms
Tverberg’s theorem states that for any k ≥ 2 and any set P ⊂ ℝ^d of at least (d + 1)(k - 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem lies in the complexity class PPAD ∩ PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm.
We present a deterministic algorithm that finds for any n-point set P ⊂ ℝ^d and any k ∈ {2, … , n} in O(nd ⌈log k⌉) time a k-partition of P such that there is a ball of radius O((k/√n)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, and that there are no approximation algorithms that are truly polynomial in any dimension, our result provides a remarkably efficient and simple new notion of approximation.
Our main contribution is to generalize Sarkaria’s method [Israel Journal Math., 1992] to reduce the Tverberg problem to the Colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.
Tverberg’s theorem
Colorful Carathéodory Theorem
Tensor lifting
Theory of computation~Computational geometry
Theory of computation~Graph algorithms analysis
31:1-31:17
Regular Paper
Supported in part by ERC StG 757609.
A full version is available on the arXiv (https://arxiv.org/abs/1907.04284).
We would like to thank Frédéric Meunier for stimulating discussions about the Colorful Carathéodory theorem and related problems and for hosting us during a research stay at his lab. We would also like to thank Sergey Bereg for helpful comments on a previous version of this manuscript.
Aruni
Choudhary
Aruni Choudhary
Institut für Informatik, Freie Universität Berlin, Takustraße 9, 14195 Berlin, Germany
https://orcid.org/0000-0002-9225-0829
Wolfgang
Mulzer
Wolfgang Mulzer
Institut für Informatik, Freie Universität Berlin, Takustraße 9, 14195 Berlin, Germany
https://orcid.org/0000-0002-1948-5840
10.4230/LIPIcs.SoCG.2020.31
Karim Adiprasito, Imre Bárány, and Nabil Mustafa. Theorems of Carathéodory, Helly, and Tverberg without dimension. In Proc. 30th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 2350-2360, 2019.
Noga Alon and Joel H. Spencer. The Probalistic method. John Wiley & Sons, 2008.
Jorge L. Arocha, Imre Bárány, Javier Bracho, Ruy Fabila Monroy, and Luis Montejano. Very colorful theorems. Discrete Comput. Geom., 42(2):42-154, 2009.
Imre Bárány. A generalization of Carathéodory’s theorem. Discrete Mathematics, 40(2-3):141-152, 1982.
Imre Bárány and David G. Larman. A colored version of Tverberg’s theorem. Journal of the London Mathematical Society, s2-45(2):314-320, 1992.
Imre Bárány and Shmuel Onn. Colourful linear programming and its relatives. Mathematics of Operations Research, 22(3):550-567, 1997.
Pavle Blagojević, Benjamin Matschke, and Günter Ziegler. Optimal bounds for the colored Tverberg problem. Journal of the European Mathematical Society, 017(4):739-754, 2015.
Aruni Choudhary and Wolfgang Mulzer. No-dimensional tverberg theorems and algorithms. CoRR, abs/1907.04284, 2019. URL: http://arxiv.org/abs/1907.04284.
http://arxiv.org/abs/1907.04284
Kenneth L. Clarkson, David Eppstein, Gary L. Miller, Carl Sturtivant, and Shang-Hua Teng. Approximating center points with iterative radon points. Internat. J. Comput. Geom. Appl., 6(3):357-377, 1996.
Aris Filos-Ratsikas and Paul W. Goldberg. The complexity of splitting necklaces and bisecting Ham sandwiches. In Proc. 51st Annu. ACM Sympos. Theory Comput. (STOC), pages 638-649, 2019.
Sariel Har-Peled and Mitchell Jones. Journey to the center of the point set. In Proc. 35th Int. Sympos. Comput. Geom. (SoCG), pages 41:1-41:14, 2019.
Jesús De Loera, Xavier Goaoc, Frédéric Meunier, and Nabil Mustafa. The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 56(3):415-511, 2019.
Jiří Matoušek, Martin Tancer, and Uli Wagner. A geometric proof of the colored Tverberg theorem. Discrete Comput. Geom., 47(2):245-265, 2012.
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 Proc. 28th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 1342-1351, 2017.
Gary L. Miller and Donald R. Sheehy. Approximate centerpoints with proofs. Comput. Geom. Theory Appl., 43(8):647-654, 2010.
Wolfgang Mulzer and Yannik Stein. Computational aspects of the Colorful Carathéodory theorem. Discrete Comput. Geom., 60(3):720-755, 2018.
Wolfgang Mulzer and Daniel Werner. Approximating Tverberg points in linear time for any fixed dimension. Discrete Comput. Geom., 50(2):520-535, 2013.
Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, editors. Algorithmic Game Theory. Cambridge University Press, 2007.
Johann Radon. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Mathematische Annalen, 83:113-115, 1921.
Jean-Pierre Roudneff. Partitions of points into simplices with k-dimensional intersection. I. The conic Tverberg’s theorem. European Journal of Combinatorics, 22(5):733-743, 2001.
Karanbir S. Sarkaria. Tverberg’s theorem via number fields. Israel Journal of Mathematics, 79(2-3):317-320, 1992.
Pablo Soberón. Equal coefficients and tolerance in coloured Tverberg partitions. Combinatorica, 35(2):235-252, 2015.
Daniel Spielman. Spectral graph theory. URL: http://www.cs.yale.edu/homes/spielman/561/.
http://www.cs.yale.edu/homes/spielman/561/
Arthur H. Stone and John W. Tukey. Generalized “Sandwich” theorems. Duke Mathematical Journal, 9(2):356-359, June 1942.
Helge Tverberg. A generalization of Radon’s theorem. Journal of the London Mathematical Society, s1-41(1):123-128, 1966.
Helge Tverberg. A generalization of Radon’s theorem II. Journal of the Australian Mathematical Society, 24(3):321-325, 1981.
Helge Tverberg and Siniša T. Vrećica. On generalizations of Radon’s theorem and the Ham-sandwich theorem. European Journal of Combinatorics, 14(3):259-264, 1993.
Rade T. Zivaljević and Siniša T. Vrećica. An extension of the Ham sandwich theorem. Bulletin of the London Mathematical Society, 22(2):183-186, 1990.
Rade T. Zivaljević and Siniša T. Vrećica. The colored Tverberg’s problem and complexes of injective functions. Journal of Combinatorial Theory, Series A., 61:309-318, 1992.
Aruni Choudhary and Wolfgang Mulzer
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