Discrete & Computational Geometry

, Volume 60, Issue 3, pp 720–755 | Cite as

Computational Aspects of the Colorful Carathéodory Theorem

  • Wolfgang Mulzer
  • Yannik SteinEmail author


Let \(C_1,\dots ,C_{d+1}\subset \mathbb {R}^d\) be \(d+1\) point sets, each containing the origin in its convex hull. We call these sets color classes, and we call a sequence \(p_1, \dots , p_{d+1}\) with \(p_i \in C_i\), for \(i = 1, \dots , d+1\), a colorful choice. The colorful Carathéodory theorem guarantees the existence of a colorful choice that also contains the origin in its convex hull. The computational complexity of finding such a colorful choice (ColorfulCarathéodory) is unknown. This is particularly interesting in the light of polynomial-time reductions from several related problems, such as computing centerpoints, to ColorfulCarathéodory. We define a novel notion of approximation that is compatible with the polynomial-time reductions to ColorfulCarathéodory: a sequence that contains at most k points from each color class is called a k-colorful choice. We present an algorithm that for any fixed \(\varepsilon > 0\), outputs an \(\lceil \varepsilon d\rceil \)-colorful choice containing the origin in its convex hull in polynomial time. Furthermore, we consider a related problem of ColorfulCarathéodory: in the nearest colorful polytope problem (Ncp), we are given sets \(C_1,\dots ,C_n\subset \mathbb {R}^d\) that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing a local optimum for Ncp is PLS-complete, while computing a global optimum is NP-hard.


Colorful Carathéodory Theorem PLS Approximation 

Mathematics Subject Classification

68Q25 68W25 68W40 



We would like to thank Frédéric Meunier and Pauline Sarrabezolles for interesting discussions on the colorful Carathéodory problem and for hosting us during multiple research stays at the École Nationale des Ponts et Chaussées. Furthermore, we would like to thank the anonymous reviewers for their detailed reading of our paper and for their helpful and encouraging comments on previous versions.


WM was supported in part by DFG Grants MU 3501/1 and MU 3501/2 and ERC StG 757609. YS was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408) and by GIF Grant 1161.


  1. 1.
    Aarts, E., Lenstra, J.K. (eds.): Local Search in Combinatorial Optimization. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
  2. 2.
    Arocha, J.L., Bárány, I., Bracho, J., Fabila, R., Montejano, L.: Very colorful theorems. Discrete Comput. Geom. 42(2), 142–154 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bárány, I., Onn, S.: Colourful linear programming and its relatives. Math. Oper. Res. 22(3), 550–567 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barman, S.: Approximating Nash equilibria and dense bipartite subgraphs via an approximate version of Carathéodory’s theorem. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pp. 361–369. ACM, New York (2015)Google Scholar
  6. 6.
    Blum, M., Pratt, V., Tarjan, R.E., Floyd, R.W., Rivest, R.L.: Time bounds for selection. J. Comput. Syst. Sci. 7(4), 448–461 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chan, T.M.: An optimal randomized algorithm for maximum Tukey depth. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’04), pp. 430–436. ACM, New York (2004)Google Scholar
  8. 8.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC’04), pp. 604–612. ACM, New York (2004)Google Scholar
  9. 9.
    Jadhav, S., Mukhopadhyay, A.: Computing a centerpoint of a finite planar set of points in linear time. Discrete Comput. Geom. 12(3), 291–312 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. Comput. System Sci. 37(1), 79–100 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kapoor, S., Vaidya, P.M.: Fast algorithms for convex quadratic programming and multicommodity flows. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC’86), pp. 147–159. ACM, New York (1986)Google Scholar
  12. 12.
    Kirchberger, P.: Über Tchebychefsche Annäherungsmethoden. Math. Ann. 57(4), 509–540 (1903)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kozlov, M.K., Tarasov, S.P., Khachiyan, L.G.: The polynomial solvability of convex quadratic programming. USSR Comput. Math. Math. Phys. 20(5), 223–228 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)CrossRefGoogle Scholar
  15. 15.
    Megiddo, N., Papadimitriou, C.H.: On total functions, existence theorems and computational complexity. Theor. Comput. Sci. 81(2), 317–324 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Meunier, F., Deza, A.: A further generalization of the colourful Carathéodory theorem. In: Bezdek, K., Deza, A., Ye, Y. (eds.) Discrete Geometry and Optimization. Fields Institute Communications, vol. 69, pp. 179–190. Springer, Berlin (2013)CrossRefGoogle Scholar
  17. 17.
    Meunier, F., Mulzer, W., Sarrabezolles, P., Stein, Y.: The rainbow at the end of the line—a PPAD formulation of the colorful Carathéodory theorem with applications. In: Proceedings of the 28th Annual ACM–SIAM Symposium on Discrete Algorithms (SODA’17), pp. 1342–1351. SIAM, Philadelphia (2017)Google Scholar
  18. 18.
    Meunier, F., Sarrabezolles, P.: Colorful linear programming, Nash equilibrium, and pivots (2014). arXiv:1409.3436
  19. 19.
    Michiels, W., Aarts, E., Korst, J.: Theoretical Aspects of Local Search. Monographs in Theoretical Computer Science. Springer, Berlin (2007)zbMATHGoogle Scholar
  20. 20.
    Miller, G.L., Sheehy, D.R.: Approximate centerpoints with proofs. Comput. Geom. 43(8), 647–654 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mulzer, W., Werner, D.: Approximating Tverberg points in linear time for any fixed dimension. Discrete Comput. Geom. 50(2), 520–535 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: The complexity of the Lin-Kernighan heuristic for the traveling salesman problem. SIAM J. Comput. 21(3), 450–465 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Preparata, F.P., Shamos, M.I.: Computational Geometry. Texts and Monographs in Computer Science. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  24. 24.
    Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291–300 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Roudneff, J.P.: Partitions of points into simplices with \(k\)-dimensional intersection. I. The conic Tverberg’s theorem. Eur. J. Comb. 22(5), 733–743 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sarkaria, K.S.: Tverberg’s theorem via number fields. Israel J. Math. 79(2–3), 317–320 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schäffer, A.A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM J. Comput. 20(1), 56–87 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Teng, S.-H.: Points, Spheres, and Separators: A Unified Geometric Approach to Graph Partitioning. Ph.D. thesis, Carnegie Mellon University (1991)Google Scholar
  29. 29.
    Tverberg, H.: A generalization of Radon’s theorem. J. Lond. Math. Soc. 41(1), 123–128 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tverberg, H.: A generalization of Radon’s theorem II. Bull. Aust. Math. Soc. 24(3), 321–325 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tverberg, H., Vrećica, S.: On generalizations of Radon’s theorem and the ham sandwich theorem. Eur. J. Comb. 14(3), 259–264 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany

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