Document Open Access Logo

Further Collapses in TFNP

Authors Mika Göös, Alexandros Hollender , Siddhartha Jain , Gilbert Maystre, William Pires, Robert Robere , Ran Tao

PDF
Thumbnail PDF

File

LIPIcs.CCC.2022.33.pdf
  • Filesize: 0.76 MB
  • 15 pages

Document Identifiers

Author Details

Mika Göös
  • EPFL, Lausanne, Switzerland
Alexandros Hollender
  • University of Oxford, UK
Siddhartha Jain
  • EPFL, Lausanne, Switzerland
Gilbert Maystre
  • EPFL, Lausanne, Switzerland
William Pires
  • McGill University, Montreal, Canada
Robert Robere
  • McGill University, Montreal, Canada
Ran Tao
  • McGill University, Montreal, Canada

Acknowledgements

We thank Aviad Rubinstein for his many questions during e-mail correspondence, and the anonymous reviewers for their suggestions that helped improve the presentation of the paper.

Cite AsGet BibTex

Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further Collapses in TFNP. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.33

Abstract

We show EOPL = PLS ∩ PPAD. Here the class EOPL consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubáček and Yogev (SICOMP 2020) and Fearnley et al. (JCSS 2020). In particular, our result yields a new simpler proof of the breakthrough collapse CLS = PLS ∩ PPAD by Fearnley et al. (STOC 2021). We also prove a companion result SOPL = PLS ∩ PPADS, where SOPL is the class associated with the Sink-of-Potential-Line problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • TFNP
  • PPAD
  • PLS
  • EOPL

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The relative complexity of NP search problems. Journal of Computer and System Sciences, 57(1):3-19, 1998. URL: https://doi.org/10.1006/jcss.1998.1575.
  2. Joshua Buresh-Oppenheim and Tsuyoshi Morioka. Relativized NP search problems and propositional proof systems. In Proceedings of the 19th IEEE Conference on Computational Complexity (CCC), pages 54-67, 2004. URL: https://doi.org/10.1109/CCC.2004.1313795.
  3. 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), pages 31:1-31:18, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.31.
  4. Constantinos Daskalakis, Paul Goldberg, and Christos Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009. URL: https://doi.org/10.1137/070699652.
  5. Constantinos Daskalakis and Christos Papadimitriou. Continuous local search. In Proceedings of the 22nd Symposium on Discrete Algorithms (SODA), pages 790-804, January 2011. URL: https://doi.org/10.1137/1.9781611973082.62.
  6. Jérôme Dohrau, Bernd Gärtner, Manuel Kohler, Jiří Matoušek, and Emo Welzl. ARRIVAL: A zero-player graph game in NP ∩ coNP. In A Journey Through Discrete Mathematics, pages 367-374. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-44479-6_14.
  7. Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski’s theorem, supermodular games, and the complexity of equilibria. In Proceedings of the 11th Innovations in Theoretical Computer Science Conference (ITCS), volume 151, pages 18:1-18:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.18.
  8. John Fearnley, Paul Goldberg, Alexandros Hollender, and Rahul Savani. The complexity of gradient descent: CLS = PPAD ∩ PLS. In Proceedings of the 53rd Symposium on Theory of Computing (STOC), pages 46-59, 2021. URL: https://doi.org/10.1145/3406325.3451052.
  9. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique end of potential line. Journal of Computer and System Sciences, 114:1-35, 2020. URL: https://doi.org/10.1016/j.jcss.2020.05.007.
  10. Bernd Gärtner, Thomas Dueholm Hansen, Pavel Hubácek, Karel Král, Hagar Mosaad, and Veronika Slívová. ARRIVAL: Next stop in CLS. In 45th International Colloquium on Automata, Languages, and Programming (ICALP), volume 107, pages 60:1-60:13. Schloss Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPICS.ICALP.2018.60.
  11. Paul Goldberg and Alexandros Hollender. The Hairy Ball problem is PPAD-complete. Journal of Computer and System Sciences, 122:34-62, 2021. URL: https://doi.org/10.1016/j.jcss.2021.05.004.
  12. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Proceedings of the 10th Innovations in Theoretical Computer Science Conference (ITCS), volume 124, pages 38:1-38:19, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  13. Alexandros Hollender. Structural results for total search complexity classes with applications to game theory and optimisation. PhD thesis, University of Oxford, 2021. URL: https://ora.ox.ac.uk/objects/uuid:67e2d80b-76bf-4b49-9b7d-8bbd91633dd7.
  14. Pavel Hubáček and Eylon Yogev. Hardness of continuous local search: Query complexity and cryptographic lower bounds. SIAM Journal on Computing, 49(6):1128-1172, 2020. URL: https://doi.org/10.1137/17m1118014.
  15. Takashi Ishizuka. The complexity of the parity argument with potential. Journal of Computer and System Sciences, 120:14-41, September 2021. URL: https://doi.org/10.1016/j.jcss.2021.03.004.
  16. David Johnson, Christos Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79-100, August 1988. URL: https://doi.org/10.1016/0022-0000(88)90046-3.
  17. Nimrod Megiddo and Christos Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, April 1991. URL: https://doi.org/10.1016/0304-3975(91)90200-L.
  18. 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 Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1342-1351, 2017. URL: https://doi.org/10.1137/1.9781611974782.87.
  19. Tsuyoshi Morioka. Classification of search problems and their definability in bounded arithmetic. Master’s thesis, University of Toronto, 2001. URL: https://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/MQ58775.pdf.
  20. Christos Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498-532, June 1994. URL: https://doi.org/10.1016/s0022-0000(05)80063-7.
  21. Patrick Schnider. The Complexity of Sharing a Pizza. In 32nd International Symposium on Algorithms and Computation (ISAAC), pages 13:1-13:15, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.13.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact