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Computational Complexity of the α-Ham-Sandwich Problem

Authors Man-Kwun Chiu , Aruni Choudhary , Wolfgang Mulzer

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Man-Kwun Chiu
  • Institut für Informatik, Freie Universität Berlin, Germany
Aruni Choudhary
  • Institut für Informatik, Freie Universität Berlin, Germany
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany

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Man-Kwun Chiu, Aruni Choudhary, and Wolfgang Mulzer. Computational Complexity of the α-Ham-Sandwich Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 31:1-31:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


The classic Ham-Sandwich theorem states that for any d measurable sets in ℝ^d, there is a hyperplane that bisects them simultaneously. An extension by Bárány, Hubard, and Jerónimo [DCG 2008] states that if the sets are convex and well-separated, then for any given α₁, … , α_d ∈ [0, 1], there is a unique oriented hyperplane that cuts off a respective fraction α₁, … , α_d from each set. Steiger and Zhao [DCG 2010] proved a discrete analogue of this theorem, which we call the α-Ham-Sandwich theorem. They gave an algorithm to find the hyperplane in time O(n (log n)^{d-3}), where n is the total number of input points. The computational complexity of this search problem in high dimensions is open, quite unlike the complexity of the Ham-Sandwich problem, which is now known to be PPA-complete (Filos-Ratsikas and Goldberg [STOC 2019]). Recently, Fearnley, Gordon, Mehta, and Savani [ICALP 2019] introduced a new sub-class of CLS (Continuous Local Search) called Unique End-of-Potential Line (UEOPL). This class captures problems in CLS that have unique solutions. We show that for the α-Ham-Sandwich theorem, the search problem of finding the dividing hyperplane lies in UEOPL. This gives the first non-trivial containment of the problem in a complexity class and places it in the company of classic search problems such as finding the fixed point of a contraction map, the unique sink orientation problem and the P-matrix linear complementarity problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Ham-Sandwich Theorem
  • Computational Complexity
  • Continuous Local Search


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  1. James Aisenberg, Maria Luisa Bonet, and Sam Buss. 2-D Tucker is PPA-complete. J. Comput. System Sci., 108:92-103, 2020. Google Scholar
  2. Noga Alon and Douglas B. West. The Borsuk-Ulam theorem and bisection of necklaces. Proc. American Mathematical Society, 98(4):623-628, 1986. Google Scholar
  3. Imre Bárány. A generalization of Carathéodory’s theorem. Discrete Mathematics, 40(2-3):141-152, 1982. Google Scholar
  4. Imre Bárány, Alfredo Hubard, and Jesús Jerónimo. Slicing convex sets and measures by a hyperplane. Discrete Comput. Geom., 39(1):67-75, 2008. Google Scholar
  5. Imre Bárány and Jiří Matoušek. Simultaneous partitions of measures by k-fans. Discrete Comput. Geom., 25(3):317-334, 2001. Google Scholar
  6. Luis Barba, Alexander Pilz, and Patrick Schnider. Sharing a pizza: bisecting masses with two cuts. CoRR, abs/1904.02502, 2019. Google Scholar
  7. Sergey Bereg. Equipartitions of measures by 2-fans. Discrete Comput. Geom., 34(1):87-96, 2005. Google Scholar
  8. Sergey Bereg. Computing generalized ham-sandwich cuts. Inform. Process. Lett., 112(13):532-534, 2012. Google Scholar
  9. Pavle V. M. Blagojević , Aleksandra Dimitrijević Blagojević , Roman Karasev, and Jonathan Kliem. More bisections by hyperplane arrangements, 2018. URL:
  10. Xi Chen, Decheng Dai, Ye Du, and Shang-Hua Teng. Settling the complexity of arrow-debreu equilibria in markets with additively separable utilities. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pages 273-282, 2009. Google Scholar
  11. Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. J. ACM, 56(3):14:1-14:57, 2009. Google Scholar
  12. Xi Chen, Dimitris Paparas, and Mihalis Yannakakis. The complexity of non-monotone markets. J. ACM, 64(3):20:1-20:56, 2017. Google Scholar
  13. Man-Kwun Chiu, Aruni Choudhary, and Wolfgang Mulzer. Computational complexity of the α-ham-sandwich problem, 2020. URL:
  14. Aruni Choudhary and Wolfgang Mulzer. No-dimensional Tverberg theorems and algorithms. In Proc. 36th Int. Sympos. Comput. Geom. (SoCG), page to appear, 2020. Google Scholar
  15. S. J. Chung. NP-completeness of the Linear Complementarity Problem. Journal of Optimization Theory and Applications, 60(3):393-399, 1989. Google Scholar
  16. Gregory E. Coxson. The P-matrix problem is co-NP-complete. Mathematical Programming, 64(1):173-178, 1994. Google Scholar
  17. Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash Equilibrium. SIAM J. Comput., 39(1):195-259, 2009. Google Scholar
  18. Constantinos Daskalakis and Christos H. Papadimitriou. Continuous Local Search. In Proc. 22nd Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 790-804, 2011. Google Scholar
  19. Vladimir L. Dol'nikov. A generalization of the Ham-sandwich theorem. Mathematical Notes, 52(2):771-779, 1992. Google Scholar
  20. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique End of Potential Line. In Proc. 46th Internat. Colloq. Automata Lang. Program. (ICALP), pages 56:1-56:15, 2019. Google Scholar
  21. 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. Google Scholar
  22. Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. Consensus-halving: Does it ever get easier? CoRR, abs/2002.11437, 2020. URL:
  23. Vincent Froese, Iyad A. Kanj, André Nichterlein, and Rolf Niedermeier. Finding points in general position. Internat. J. Comput. Geom. Appl., 27(4):277-296, 2017. Google Scholar
  24. Paul W. Goldberg and Alexandros Hollender. The Hairy Ball problem is PPAD-complete. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, pages 65:1-65:14, 2019. Google Scholar
  25. Alfredo Hubard and Roman Karasev. Bisecting measures with hyperplane arrangements. Mathematical Proceedings of the Cambridge Philosophical Society, pages 1-9, 2019. Google Scholar
  26. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? J. Comput. System Sci., 37(1):79-100, 1988. Google Scholar
  27. Roman N. Karasev. Topological methods in combinatorial geometry. Russian Mathematical Surveys, 63(6):1031-1078, 2008. Google Scholar
  28. Chi-Yuan Lo, Jiří Matoušek, and William L. Steiger. Algorithms for Ham-sandwich cuts. Discrete Comput. Geom., 11:433-452, 1994. Google Scholar
  29. 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. Google Scholar
  30. Jiří Matoušek. Using the Borsuk-Ulam theorem. Springer-Verlag Berlin Heidelberg, 2003. Google Scholar
  31. 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. Google Scholar
  32. Wolfgang Mulzer and Yannik Stein. Computational aspects of the Colorful carathéodory theorem. Discrete Comput. Geom., 60(3):720-755, 2018. Google Scholar
  33. Abraham Othman, Christos H. Papadimitriou, and Aviad Rubinstein. The complexity of fairness through equilibrium. ACM Trans. Economics and Comput., 4(4):20:1-20:19, 2016. Google Scholar
  34. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. System Sci., 48(3):498-532, 1994. Google Scholar
  35. Richard Rado. A theorem on general measure. Journal of the London Mathematical Society, s1-21(4):291-300, 1946. Google Scholar
  36. Karanbir S. Sarkaria. Tverberg’s theorem via number fields. Israel Journal of Mathematics, 79(2-3):317-320, 1992. Google Scholar
  37. Patrick Schnider. Equipartitions with wedges and cones. CoRR, abs/1910.13352, 2019. Google Scholar
  38. Patrick Schnider. Ham-sandwich cuts and center transversals in subspaces. In Proc. 35th Int. Sympos. Comput. Geom. (SoCG), pages 56:1-56:15, 2019. Google Scholar
  39. Forest W. Simmons and Francis Edward Su. Consensus-halving via theorems of Borsuk-Ulam and Tucker. Mathematical Social Sciences, 45(1):15-25, 2003. Google Scholar
  40. William Steiger and Jihui Zhao. Generalized Ham-sandwich cuts. Discrete Comput. Geom., 44(3):535-545, 2010. Google Scholar
  41. Arthur H. Stone and John W. Tukey. Generalized “Sandwich” theorems. Duke Mathematical Journal, 9(2):356-359, June 1942. Google Scholar
  42. Albert W. Tucker. Some topological properties of disk and sphere. In Proc. First Canadian Math. Congress, Montreal, 1945, pages 285-309. University of Toronto Press, Toronto, 1946. Google Scholar
  43. Helge Tverberg. A generalization of Radon’s theorem. Journal of the London Mathematical Society, s1-41(1):123-128, 1966. Google Scholar
  44. 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. Google Scholar
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