Extremal Combinatorics, Iterated Pigeonhole Arguments and Generalizations of PPP

Authors Amol Pasarkar , Christos Papadimitriou, Mihalis Yannakakis



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2023.88.pdf
  • Filesize: 0.77 MB
  • 20 pages

Document Identifiers

Author Details

Amol Pasarkar
  • Columbia University, New York, NY, USA
Christos Papadimitriou
  • Columbia University, New York, NY, USA
Mihalis Yannakakis
  • Columbia University, New York, NY, USA

Cite AsGet BibTex

Amol Pasarkar, Christos Papadimitriou, and Mihalis Yannakakis. Extremal Combinatorics, Iterated Pigeonhole Arguments and Generalizations of PPP. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 88:1-88:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.88

Abstract

We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey’s theorem, the Sunflower theorem, the Erdős-Ko-Rado lemma, and König’s lemma. Whether the first two of these four problems are PLC-complete is an important open question which we pursue; in contrast, we show that the latter two are PPP-complete. Finally, we reframe PPP as an optimization problem, and define a hierarchy of such problems related to Turàn’s theorem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
Keywords
  • Total Complexity
  • Extremal Combinatorics
  • Pigeonhole Principle

Metrics

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

References

  1. Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Improved bounds for the sunflower lemma. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 624-630, 2020. Google Scholar
  2. Bèla Bollobàs. Extremal Graph Theory. Dover, 2013. Google Scholar
  3. David Conlon and Asaf Ferber. Lower bounds for multicolor ramsey numbers, 2020. URL: https://doi.org/10.48550/arXiv.2009.10458.
  4. Michel Deza and Peter Frankl. Every large set of equidistant (0, +1, −1)-vectors forms a sunflower. Combinatorica, 1:225-231, September 1981. Google Scholar
  5. P. Erdös and R. Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, s1-35(1):85-90, 1960. Google Scholar
  6. Paul W. Goldberg and Christos H. Papadimitriou. Towards a unified complexity theory of total functions. Journal of Computer and System Sciences, 94:167-192, 2018. Google Scholar
  7. Ronald L Graham, Bruce L Rothschild, and Joel H Spencer. Ramsey theory, volume 20. 'John Wiley & Sons', 1990. Google Scholar
  8. Emil Jeřábek. Integer factoring and modular square roots. Journal of Computer and System Sciences, 82(2):380-394, 2016. Google Scholar
  9. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? Journal of Computer and System Sciences, 37(1):79-100, 1988. Google Scholar
  10. Stasys Jukna. Extremal Combinatorics With Applications in Computer Science. Springer Berlin, 2013. Google Scholar
  11. Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos H. Papadimitriou. Total functions in the polynomial hierarchy. In 12th Innovations in Theoretical Computer Science Conference, ITCS, volume 185 of LIPIcs, pages 44:1-44:18, 2021. Google Scholar
  12. Ilan Komargodski, Moni Naor, and Eylon Yogev. White-box vs. black-box complexity of search problems: Ramsey and graph property testing. J. ACM, 66(5), July 2019. Google Scholar
  13. Nimrod Megiddo and Christos H. Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, 1991. Google Scholar
  14. 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
  15. Katerina Sotiraki, Manolis Zampetakis, and Giorgos Zirdelis. Ppp-completeness with connections to cryptography. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 148-158. IEEE Computer Society, 2018. Google Scholar