ABSTRACT
In the ε-Consensus-Halving problem, we are given n probability measures v1, …, vn on the interval R = [0,1], and the goal is to partition R into two parts R+ and R− using at most n cuts, so that |vi(R+) − vi(R−)| ≤ ε for all i. This fundamental fair division problem was the first natural problem shown to be complete for the class PPA, and all subsequent PPA-completeness results for other natural problems have been obtained by reducing from it.
We show that ε-Consensus-Halving is PPA-complete even when the parameter ε is a constant. In fact, we prove that this holds for any constant ε < 1/5. As a result, we obtain constant inapproximability results for all known natural PPA-complete problems, including Necklace-Splitting, the Discrete-Ham-Sandwich problem, two variants of the pizza sharing problem, and for finding fair independent sets in cycles and paths.
- Ron Aharoni, Noga Alon, Eli Berger, Maria Chudnovsky, Dani Kotlar, Martin Loebl, and Ran Ziv. 2017. Fair representation by independent sets. In A Journey Through Discrete Mathematics. Springer, 31–58. https://doi.org/10.1007/978-3-319-44479-6_2 Google ScholarCross Ref
- James Aisenberg, Maria Luisa Bonet, and Sam Buss. 2020. 2-D Tucker is PPA complete. J. Comput. System Sci., 108 (2020), 92–103. https://doi.org/10.1016/j.jcss.2019.09.002 Google ScholarCross Ref
- Meysam Alishahi and Frédéric Meunier. 2017. Fair splitting of colored paths. The Electronic Journal of Combinatorics, 24, 3 (2017), P3.41. https://doi.org/10.37236/7079 Google ScholarCross Ref
- Noga Alon. 1987. Splitting necklaces. Advances in Mathematics, 63, 3 (1987), 247–253. https://doi.org/10.1016/0001-8708(87)90055-7 Google ScholarCross Ref
- Noga Alon and Andrei Graur. 2021. Efficient Splitting of Necklaces. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP). 14:1–14:17. https://doi.org/10.4230/LIPIcs.ICALP.2021.14 Google ScholarCross Ref
- Noga Alon and Douglas B. West. 1986. The Borsuk-Ulam Theorem and Bisection of Necklaces. Proc. Amer. Math. Soc., 98, 4 (1986), 623–628. https://doi.org/10.2307/2045739 Google ScholarCross Ref
- Luis Barba, Alexander Pilz, and Patrick Schnider. 2019. Sharing a pizza: bisecting masses with two cuts. arxiv:1904.02502.Google Scholar
- Eleni Batziou, Kristoffer Arnsfelt Hansen, and Kasper Høgh. 2021. Strong Approximate Consensus Halving and the Borsuk-Ulam Theorem. In Proceedings of the 48th International Colloquium on Automata, Languages, and Programming (ICALP). 24:1–24:20. https://doi.org/10.4230/LIPIcs.ICALP.2021.24 Google ScholarCross Ref
- Alexander Black, Umur Cetin, Florian Frick, Alexander Pacun, and Linus Setiabrata. 2020. Fair splittings by independent sets in sparse graphs. Israel Journal of Mathematics, 236 (2020), 603–627. https://doi.org/10.1007/s11856-020-1980-5 Google ScholarCross Ref
- Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 2009. Settling the complexity of computing two-player Nash equilibria. J. ACM, 56, 3 (2009), 14:1–14:57. https://doi.org/10.1145/1516512.1516516 Google ScholarDigital Library
- Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. 2009. The complexity of computing a Nash equilibrium. SIAM J. Comput., 39, 1 (2009), 195–259. https://doi.org/10.1137/070699652 Google ScholarDigital Library
- Argyrios Deligkas, John Fearnley, and Themistoklis Melissourgos. 2020. Pizza Sharing is PPA-hard. arxiv:2012.14236.Google Scholar
- Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis. 2021. Computing exact solutions of consensus halving and the Borsuk-Ulam theorem. J. Comput. System Sci., 117 (2021), 75–98. https://doi.org/10.1016/j.jcss.2020.10.006 Google ScholarCross Ref
- Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis. 2022. Approximating the existential theory of the reals. J. Comput. System Sci., 125 (2022), 106–128. https://doi.org/10.1016/j.jcss.2021.11.002 Google ScholarDigital Library
- Argyrios Deligkas, Aris Filos-Ratsikas, and Alexandros Hollender. 2021. Two’s Company, Three’s a Crowd: Consensus-Halving for a Constant Number of Agents. In Proceedings of the 22nd ACM Conference on Economics and Computation (EC). 347–368. https://doi.org/10.1145/3465456.3467625 Google ScholarDigital Library
- Xiaotie Deng, Zhe Feng, and Rucha Kulkarni. 2017. Octahedral Tucker is PPA-complete. Electronic Colloquium on Computational Complexity (ECCC). https://eccc.weizmann.ac.il/report/2017/118/Google Scholar
- Kousha Etessami and Mihalis Yannakakis. 2010. On the Complexity of Nash Equilibria and Other Fixed Points. SIAM J. Comput., 39, 6 (2010), 2531–2597. https://doi.org/10.1137/080720826 Google ScholarDigital Library
- Aris Filos-Ratsikas, Søren Kristoffer Still Frederiksen, Paul W. Goldberg, and Jie Zhang. 2018. Hardness Results for Consensus-Halving. In Proceedings of the 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS). 24:1–24:16. https://doi.org/10.4230/LIPIcs.MFCS.2018.24 Google ScholarCross Ref
- Aris Filos-Ratsikas and Paul W. Goldberg. 2018. Consensus Halving is PPA-complete. In Proceedings of the 50th ACM Symposium on Theory of Computing (STOC). 51–64. https://doi.org/10.1145/3188745.3188880 Google ScholarDigital Library
- Aris Filos-Ratsikas and Paul W. Goldberg. 2019. The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches. In Proceedings of the 51st ACM Symposium on Theory of Computing (STOC). 638–649. https://doi.org/10.1145/3313276.3316334 Google ScholarDigital Library
- Aris Filos-Ratsikas and Paul W. Goldberg. 2022. The Complexity of Necklace Splitting, Consensus-Halving, and Discrete Ham Sandwich. SIAM J. Comput., https://doi.org/10.1137/20m1312678 Google ScholarCross Ref
- Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. 2020. Consensus-Halving: Does it Ever Get Easier? In Proceedings of the 21st ACM Conference on Economics and Computation (EC). 381–399. https://doi.org/10.1145/3391403.3399527 Google ScholarDigital Library
- Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. 2021. A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting. In Proceedings of the 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA). 2615–2634. https://doi.org/10.1137/1.9781611976465.155 Google ScholarCross Ref
- Charles H. Goldberg and Douglas B. West. 1985. Bisection of Circle Colorings. SIAM Journal on Algebraic Discrete Methods, 6, 1 (1985), 93–106. https://doi.org/10.1137/0606010 Google ScholarDigital Library
- Paul W. Goldberg, Alexandros Hollender, Ayumi Igarashi, Pasin Manurangsi, and Warut Suksompong. 2022. Consensus Halving for Sets of Items. Mathematics of Operations Research, https://doi.org/10.1287/moor.2021.1249 Google ScholarCross Ref
- Ishay Haviv. 2021. The Complexity of Finding Fair Independent Sets in Cycles. In Proceedings of the 12th Innovations in Theoretical Computer Science Conference (ITCS). 4:1–4:14. https://doi.org/10.4230/LIPIcs.ITCS.2021.4 Google ScholarCross Ref
- Charles R. Hobby and John R. Rice. 1965. A moment problem in L1 approximation. Proc. Amer. Math. Soc., 16, 4 (1965), 665–670. https://doi.org/10.2307/2033900 Google ScholarCross Ref
- Alexandros Hollender. 2021. The classes PPA-k: Existence from arguments modulo k. Theoretical Computer Science, 885 (2021), 15–29. https://doi.org/10.1016/j.tcs.2021.06.016 Google ScholarDigital Library
- Alfredo Hubard and Roman Karasev. 2020. Bisecting measures with hyperplane arrangements. Mathematical Proceedings of the Cambridge Philosophical Society, 169, 3 (2020), 639–647. https://doi.org/10.1017/S0305004119000380 Google ScholarCross Ref
- Roman N. Karasev, Edgardo Roldán-Pensado, and Pablo Soberón. 2016. Measure partitions using hyperplanes with fixed directions. Israel Journal of Mathematics, 212 (2016), 705–728. https://doi.org/10.1007/s11856-016-1303-z Google ScholarCross Ref
- Nimrod Megiddo and Christos H. Papadimitriou. 1991. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81, 2 (1991), 317–324. https://doi.org/10.1016/0304-3975(91)90200-L Google ScholarDigital Library
- Christos H. Papadimitriou. 1994. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. System Sci., 48, 3 (1994), 498–532. https://doi.org/10.1016/S0022-0000(05)80063-7 Google ScholarDigital Library
- Aviad Rubinstein. 2018. Inapproximability of Nash equilibrium. SIAM J. Comput., 47, 3 (2018), 917–959. https://doi.org/10.1137/15M1039274 Google ScholarCross Ref
- Patrick Schnider. 2021. The Complexity of Sharing a Pizza. In Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC). 13:1–13:15. https://doi.org/10.4230/LIPIcs.ISAAC.2021.13 Google ScholarCross Ref
- Forest W. Simmons and Francis E. Su. 2003. Consensus-halving via theorems of Borsuk-Ulam and Tucker. Mathematical social sciences, 45, 1 (2003), 15–25. https://doi.org/10.1016/S0165-4896(02)00087-2 Google ScholarCross Ref
- Albert W. Tucker. 1945. Some Topological Properties of Disk and Sphere. In Proceedings of the First Canadian Math. Congress, Montreal. University of Toronto Press, 286–309.Google Scholar
Index Terms
- Constant inapproximability for PPA
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