Consensus Division in an Arbitrary Ratio

Authors Paul Goldberg , Jiawei Li



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Author Details

Paul Goldberg
  • University of Oxford, UK
Jiawei Li
  • The University of Texas at Austin, TX, USA

Acknowledgements

We thank Alexandros Hollender for pointing out the paper [Stromquist and Woodall, 1985] to us. We thank the anonymous ITCS reviewers for their helpful comments. Jiawei Li wants to thank Xiaotie Deng for introducing the consensus-halving problem to him.

Cite AsGet BibTex

Paul Goldberg and Jiawei Li. Consensus Division in an Arbitrary Ratio. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.57

Abstract

We consider the problem of partitioning a line segment into two subsets, so that n finite measures all have the same ratio of values for the subsets. Letting α ∈ [0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which α = 1/2. Stromquist and Woodall [Stromquist and Woodall, 1985] showed that for any α, there exists a solution using 2n cuts of the segment. They also showed that if α is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values α. For α = 𝓁/k, we show a lower bound of (k-1)/k ⋅ 2n - O(1) cuts; we also obtain almost matching upper bounds for a large subset of rational α. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1) when using the minimal number of cuts for each instance is required, the problem is NP-hard for any α; 2) for a large subset of rational α = 𝓁/k, when (k-1)/k ⋅ 2n cuts are available, the problem is in PPA-k under Turing reduction; 3) when 2n cuts are allowed, the problem belongs to PPA for any α; more generally, the problem belong to PPA-p for any prime p if 2(p-1)⋅⌈p/2⌉/⌊p/2⌋ ⋅ n cuts are available.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Consensus Halving
  • TFNP
  • PPA-k
  • Necklace Splitting

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References

  1. Noga Alon. Splitting necklaces. Advances in Mathematics, 63(3):247-253, 1987. Google Scholar
  2. Noga Alon and Andrei Graur. Efficient splitting of measures and necklaces. CoRR, abs/2006.16613, 2020. URL: http://arxiv.org/abs/2006.16613.
  3. Noga Alon and Douglas B West. The borsuk-ulam theorem and bisection of necklaces. Proceedings of the American Mathematical Society, 98(4):623-628, 1986. Google Scholar
  4. Georgios Amanatidis, Haris Aziz, Georgios Birmpas, Aris Filos-Ratsikas, Bo Li, Hervé Moulin, Alexandros A. Voudouris, and Xiaowei Wu. Fair division of indivisible goods: A survey. CoRR, abs/2208.08782, 2022. URL: https://doi.org/10.48550/arXiv.2208.08782.
  5. Haris Aziz, Hervé Moulin, and Fedor Sandomirskiy. A polynomial-time algorithm for computing a pareto optimal and almost proportional allocation. Oper. Res. Lett., 48(5):573-578, 2020. URL: https://doi.org/10.1016/j.orl.2020.07.005.
  6. Eleni Batziou, Kristoffer Arnsfelt Hansen, and Kasper Høgh. Strong approximate consensus halving and the borsuk-ulam theorem. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.24.
  7. Mithun Chakraborty, Ayumi Igarashi, Warut Suksompong, and Yair Zick. Weighted envy-freeness in indivisible item allocation. ACM Trans. Economics and Comput., 9(3):18:1-18:39, 2021. URL: https://doi.org/10.1145/3457166.
  8. Mithun Chakraborty, Ulrike Schmidt-Kraepelin, and Warut Suksompong. Picking sequences and monotonicity in weighted fair division. Artif. Intell., 301:103578, 2021. URL: https://doi.org/10.1016/j.artint.2021.103578.
  9. Logan Crew, Bhargav Narayanan, and Sophie Spirkl. Disproportionate division. Bull. London Math. Soc., 52:885-890, 2020. Google Scholar
  10. Argyrios Deligkas, John Fearnley, Alexandros Hollender, and Themistoklis Melissourgos. Constant inapproximability for PPA. CoRR, abs/2201.10011, 2022. URL: http://arxiv.org/abs/2201.10011.
  11. Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, and Paul G. Spirakis. Computing exact solutions of consensus halving and the borsuk-ulam theorem. J. Comput. Syst. Sci., 117:75-98, 2021. URL: https://doi.org/10.1016/j.jcss.2020.10.006.
  12. Argyrios Deligkas, Aris Filos-Ratsikas, and Alexandros Hollender. Two’s company, three’s a crowd: Consensus-halving for a constant number of agents. In EC '21: The 22nd ACM Conference on Economics and Computation, pages 347-368. ACM, 2021. Google Scholar
  13. Argyrios Deligkas, Aris Filos-Ratsikas, and Alexandros Hollender. Two’s company, three’s a crowd: Consensus-halving for a constant number of agents. Artif. Intell., 313:103784, 2022. URL: https://doi.org/10.1016/j.artint.2022.103784.
  14. Kousha Etessami and Mihalis Yannakakis. On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing, 39(6):2531-2597, 2010. Google Scholar
  15. Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, and Jie Zhang. Hardness results for consensus-halving. In 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS, volume 117 of LIPIcs, pages 24:1-24:16, 2018. Google Scholar
  16. Aris Filos-Ratsikas and Paul W Goldberg. Consensus halving is PPA-complete. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 51-64, 2018. Google Scholar
  17. Aris Filos-Ratsikas and Paul W Goldberg. The complexity of splitting necklaces and bisecting ham sandwiches. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 638-649, 2019. Google Scholar
  18. Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. Consensus-halving: Does it ever get easier? In EC '20: The 21st ACM Conference on Economics and Computation, pages 381-399. ACM, 2020. Google Scholar
  19. Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, and Manolis Zampetakis. A topological characterization of modulo-p arguments and implications for necklace splitting. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, pages 2615-2634. SIAM, 2021. Google Scholar
  20. Paul W. Goldberg, Alexandros Hollender, Ayumi Igarashi, Pasin Manurangsi, and Warut Suksompong. Consensus halving for sets of items. In Web and Internet Economics - 16th International Conference, WINE, volume 12495 of LNCS, pages 384-397. Springer, 2020. Google Scholar
  21. Mika Göös, Pritish Kamath, Katerina Sotiraki, and Manolis Zampetakis. On the complexity of modulo-q arguments and the Chevalley-Warning theorem. In 35th Computational Complexity Conference, CCC, volume 169 of LIPIcs, pages 19:1-19:42. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  22. Charles R. Hobby and John R. Rice. A moment problem in l₁ approximation. Proceedings of the American Mathematical Society, 16(4):665-670, August 1965. Google Scholar
  23. Alexandros Hollender. The classes PPA-k: Existence from arguments modulo k. Theor. Comput. Sci., 885:15-29, 2021. URL: https://doi.org/10.1016/j.tcs.2021.06.016.
  24. Nimrod Megiddo and Christos H Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, 1991. Google Scholar
  25. Christos H Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and system Sciences, 48(3):498-532, 1994. Google Scholar
  26. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 216-226. ACM, 1978. URL: https://doi.org/10.1145/800133.804350.
  27. Erel Segal-Halevi. Cake-cutting with different entitlements: How many cuts are needed? J. Math. Anal. Appl., 480:1-5, 2019. Google Scholar
  28. Erel Segal-Halevi. Fair multi-cake cutting. Discret. Appl. Math., 291:15-35, 2021. Google Scholar
  29. 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
  30. Walter Stromquist and Douglas R Woodall. Sets on which several measures agree. Journal of mathematical analysis and applications, 108(1):241-248, 1985. Google Scholar
  31. Biaoshuai Tao. On existence of truthful fair cake cutting mechanisms. CoRR, abs/2104.07387, 2021. URL: http://arxiv.org/abs/2104.07387.
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