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Multiparty Communication Complexity of Collision-Finding and Cutting Planes Proofs of Concise Pigeonhole Principles

Authors Paul Beame , Michael Whitmeyer

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  • Mathematics of computing
Keywords
  • Proof Complexity
  • Communication Complexity

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Abstract

We prove several results concerning the communication complexity of a collision-finding problem, each of which has applications to the complexity of cutting-plane proofs, which make inferences based on integer linear inequalities.
In particular, we prove an Ω(n^{1-1/k} log k /2^k) lower bound on the k-party number-in-hand communication complexity of collision-finding. This implies a 2^{n^{1-o(1)}} lower bound on the size of tree-like cutting-planes refutations of the bit pigeonhole principle CNFs, which are compact and natural propositional encodings of the negation of the pigeonhole principle, improving on the best previous lower bound of 2^{Ω(√n)}. Using the method of density-restoring partitions, we also extend that previous lower bound to the full range of pigeonhole parameters.
Finally, using a refinement of a bottleneck-counting framework of Haken and Cook and Sokolov for DAG-like communication protocols, we give a 2^{Ω(n^{1/4})} lower bound on the size of fully general (not necessarily tree-like) cutting planes refutations of the same bit pigeonhole principle formulas, improving on the best previous lower bound of 2^{Ω(n^{1/8})}.

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Paul Beame and Michael Whitmeyer. Multiparty Communication Complexity of Collision-Finding and Cutting Planes Proofs of Concise Pigeonhole Principles. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.21

Author Details

Paul Beame
  • University of Washington, Seattle, WA, USA
Michael Whitmeyer
  • University of Washington, Seattle, WA, USA

Funding

Research supported by NSF grants CCF-2006359 and CCF-2422205.

References

  1. Albert Atserias, Moritz Müller, and Sergi Oliva. Lower bounds for DNF-refutations of a relativized weak pigeonhole principle. J. Symb. Log., 80(2):450-476, 2015. URL: https://doi.org/10.1017/JSL.2014.56.
  2. Paul Beame, Toniann Pitassi, and Nathan Segerlind. Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput., 37(3):845-869, 2007. URL: https://doi.org/10.1137/060654645.
  3. Paul Beame and Michael Whitmeyer. Multiparty communication complexity of collision-finding and cutting planes proofs of concise pigeonhole principles. Electron. Colloquium Comput. Complex., TR25-057, 2025. URL: https://eccc.weizmann.ac.il/report/2025/057.
  4. Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing (STOC 1995), pages 575-584, Las Vegas, Nevada, 1995. ACM. URL: https://doi.org/10.1145/225058.225275.
  5. Mark Braverman and Rotem Oshman. The communication complexity of number-in-hand set disjointness with no promise. Electron. Colloquium Comput. Complex., TR15-002, 2015. URL: https://eccc.weizmann.ac.il/report/2015/002.
  6. Samuel R. Buss and György Turán. Resolution proofs of generalized pigeonhole principles. Theor. Comput. Sci., 62(3):311-317, 1988. URL: https://doi.org/10.1016/0304-3975(88)90072-2.
  7. Arkadev Chattopadhyay and Toniann Pitassi. The story of set disjointness. SIGACT News, 41(3):59-85, 2010. URL: https://doi.org/10.1145/1855118.1855133.
  8. Stefan S. Dantchev, Nicola Galesi, Abdul Ghani, and Barnaby Martin. Proof complexity and the binary encoding of combinatorial principles. SIAM J. Comput., 53(3):764-802, 2024. URL: https://doi.org/10.1137/20M134784X.
  9. Stephan Gocht, Jakob Nordström, and Amir Yehudayoff. On division versus saturation in pseudo-boolean solving. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, pages 1711-1718. ijcai.org, August 2019. URL: https://doi.org/10.24963/IJCAI.2019/237.
  10. Mika Göös and Siddhartha Jain. Communication complexity of collision. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022, volume 245 of LIPIcs, pages 19:1-19:9. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2022.19.
  11. Dima Grigoriev, Edward A. Hirsch, and Dmitrii V. Pasechnik. Complexity of semi-algebraic proofs. In STACS 2002, 19th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings, volume 2285 of Lecture Notes in Computer Science, pages 419-430. Springer, 2002. URL: https://doi.org/10.1007/3-540-45841-7_34.
  12. Armin Haken. The intractability of resolution. Theor. Comput. Sci., 39:297-308, 1985. URL: https://doi.org/10.1016/0304-3975(85)90144-6.
  13. Armin Haken and Stephen A. Cook. An exponential lower bound for the size of monotone real circuits. J. Comput. Syst. Sci., 58(2):326-335, 1999. URL: https://doi.org/10.1006/JCSS.1998.1617.
  14. Johan Håstad. On small-depth Frege proofs for PHP. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, pages 37-49. IEEE, November 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00010.
  15. Pavel Hrubes and Pavel Pudlák. A note on monotone real circuits. Electron. Colloquium Comput. Complex., TR17-048, 2017. URL: https://eccc.weizmann.ac.il/report/2017/048.
  16. Pavel Hrubes and Pavel Pudlák. Random formulas, monotone circuits, and interpolation. Electron. Colloquium Comput. Complex., TR17-042, 2017. URL: https://eccc.weizmann.ac.il/report/2017/042.
  17. Russell Impagliazzo, Toniann Pitassi, and Alasdair Urquhart. Upper and lower bounds for tree-like cutting planes proofs. In Proceedings of the Ninth Annual Symposium on Logic in Computer Science (LICS '94), pages 220-228, Paris, France, 1994. IEEE Computer Society. URL: https://doi.org/10.1109/LICS.1994.316069.
  18. Dmitry Itsykson and Artur Riazanov. Proof complexity of natural formulas via communication arguments. In Proceeedings of the 36th Computational Complexity Conference, CCC 2021, volume 200 of LIPIcs, pages 3:1-3:34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.CCC.2021.3.
  19. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
  20. Jan Krajícek, Pavel Pudlák, and Alan R. Woods. An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Struct. Algorithms, 7(1):15-40, 1995. URL: https://doi.org/10.1002/RSA.3240070103.
  21. Saburo Muroga, Iwao Toda, and Satoru Takasu. Theory of majority decision elements. Journal of the Franklin Institute, 271(5):376-418, 1961. Google Scholar
  22. Noam Nisan. The communication complexity of threshold gates. Combinatorics, Paul Erdos is Eighty, 1(301-315):6, 1993. Google Scholar
  23. Toniann Pitassi, Paul Beame, and Russell Impagliazzo. Exponential lower bounds for the pigeonhole principle. Comput. Complex., 3:97-140, 1993. URL: https://doi.org/10.1007/BF01200117.
  24. Pavel Pudlák. Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log., 62(3):981-998, 1997. URL: https://doi.org/10.2307/2275583.
  25. Anup Rao and Amir Yehudayoff. Communication Complexity: and Applications. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108671644.
  26. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. J. ACM, 39(3):736-744, 1992. URL: https://doi.org/10.1145/146637.146684.
  27. Hanif D. Sherali and Warren P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math., 3(3):411-430, 1990. URL: https://doi.org/10.1137/0403036.
  28. Dmitry Sokolov. Dag-like communication and its applications. In Computer Science - Theory and Applications - 12th International Computer Science Symposium in Russia, CSR 2017, Proceedings, volume 10304 of Lecture Notes in Computer Science, pages 294-307. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-58747-9_26.
  29. Dmitry Sokolov. Random (log n)-CNF are hard for cutting planes (again). In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 2008-2015. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649636.
  30. Emanuele Viola. The communication complexity of addition. Combinatorica, 35(6):703-747, 2015. URL: https://doi.org/10.1007/S00493-014-3078-3.
  31. Shuo Wang, Guangxu Yang, and Jiapeng Zhang. Communication complexity of set-intersection problems and its applications. Electron. Colloquium Comput. Complex., TR23-164, 2023. URL: https://eccc.weizmann.ac.il/report/2023/164.
  32. Guangxu Yang and Jiapeng Zhang. Communication lower bounds for collision problems via density increment arguments. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 630-639. ACM, June 2024. URL: https://doi.org/10.1145/3618260.3649607.
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