Document Open Access Logo

Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication

Authors Artur Riazanov , Anastasia Sofronova , Dmitry Sokolov , Weiqiang Yuan

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.64.pdf
  • Filesize: 0.87 MB
  • 17 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • communication complexity
  • proof complexity
  • random CNF

Metrics

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

Abstract

We show that for a randomly sampled unsatisfiable O(log n)-CNF over n variables the randomized two-party communication cost of finding a clause falsified by the given variable assignment is linear in n.

Cite As Get BibTex

Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov, and Weiqiang Yuan. Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.64

Author Details

Artur Riazanov
  • EPFL, Lausanne, Switzerland
Anastasia Sofronova
  • EPFL, Lausanne, Switzerland
Dmitry Sokolov
  • EPFL, Lausanne, Switzerland
Weiqiang Yuan
  • EPFL, Lausanne, Switzerland

Funding

This project was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.

References

  1. Michael Alekhnovich. Lower bounds for k-dnf resolution on random 3-cnfs. Comput. Complex., 20(4):597-614, 2011. URL: https://doi.org/10.1007/s00037-011-0026-0.
  2. Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi Wigderson. Pseudorandom generators in propositional proof complexity. SIAM J. Comput., 34(1):67-88, 2004. URL: https://doi.org/10.1137/S0097539701389944.
  3. Michael Alekhnovich and Alexander A. Razborov. Lower bounds for polynomial calculus: Non-binomial case. Proceedings of the Steklov Institute of Mathematics, 242:18-35, 2003. Available at http://people.cs.uchicago.edu/~razborov/files/misha.pdf. Preliminary version in FOCS '01.
  4. Paul Beame, Richard M. Karp, Toniann Pitassi, and Michael E. Saks. The efficiency of resolution and davis-putnam procedures. SIAM J. Comput., 31(4):1048-1075, 2002. URL: https://doi.org/10.1137/S0097539700369156.
  5. Paul Beame, Toniann Pitassi, and Nathan Segerlind. Lower bounds for lov[a-acute]sz-schrijver systems and beyond follow from multiparty communication complexity. SIAM J. Comput., 37(3):845-869, 2007. URL: https://doi.org/10.1137/060654645.
  6. Paul Beame and Michael Whitmeyer. Multiparty Communication Complexity of Collision-Finding and Cutting Planes Proofs of Concise Pigeonhole Principles. In Keren Censor-Hillel, Fabrizio Grandoni, Joël Ouaknine, and Gabriele Puppis, editors, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025), volume 334 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:20, Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2025.21.
  7. Vašek Chvátal and Endre Szemerédi. Many hard examples for resolution. J. ACM, 35(4):759-768, October 1988. URL: https://doi.org/10.1145/48014.48016.
  8. Amit Daniely and Shai Shalev-Shwartz. Complexity theoretic limitations on learning dnf’s. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 815-830, Columbia University, New York, New York, USA, 23-26 June 2016. PMLR. URL: https://proceedings.mlr.press/v49/daniely16.html.
  9. Susanna F. de Rezende, Noah Fleming, Duri Andrea Janett, Jakob Nordström, and Shuo Pang. Truly supercritical trade-offs for resolution, cutting planes, monotone circuits, and weisfeiler–leman. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC '25, pages 1371-1382, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3717823.3718271.
  10. Susanna F. de Rezende, Jakob Nordström, and Marc Vinyals. How limited interaction hinders real communication (and what it means for proof and circuit complexity). In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 295-304. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.40.
  11. Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings of the 17th Annual IEEE Conference on Computational Complexity, Montréal, Québec, Canada, May 21-24, 2002, page 5. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/CCC.2002.10006.
  12. Noah Fleming, Denis Pankratov, Toniann Pitassi, and Robert Robere. Random Θ(log n)-CNFs are Hard for Cutting Planes. J. ACM, 69(3):19:1-19:32, 2022. URL: https://doi.org/10.1145/3486680.
  13. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. Theory Comput., 16:1-30, 2020. URL: https://doi.org/10.4086/TOC.2020.V016A013.
  14. Mika Göös, Tom Gur, Siddhartha Jain, and Jiawei Li. Quantum communication advantage in tfnp. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC '25, pages 1465-1475, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3717823.3718155.
  15. Mika Göös, Rahul Jain, and Thomas Watson. Extension complexity of independent set polytopes. SIAM J. Comput., 47(1):241-269, 2018. URL: https://doi.org/10.1137/16M109884X.
  16. Mika Göös, T. S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication versus partition number. ACM Trans. Comput. Theory, 10(1):4:1-4:20, 2018. URL: https://doi.org/10.1145/3170711.
  17. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 38:1-38:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  18. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM J. Comput., 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  19. Mika Göös, Gilbert Maystre, Kilian Risse, and Dmitry Sokolov. Supercritical tradeoffs for monotone circuits. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC '25, pages 1359-1370, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3717823.3718229.
  20. Mika Göös, Ilan Newman, Artur Riazanov, and Dmitry Sokolov. Hardness condensation by restriction. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 2016-2027. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649711.
  21. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM Journal on Computing, 47(5):1778-1806, 2018. URL: https://doi.org/10.1137/16M1082007.
  22. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for BPP. SIAM Journal on Computing, 49(4), 2020. URL: https://doi.org/10.1137/17M115339X.
  23. Dima Grigoriev. Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1):613-622, 2001. URL: https://doi.org/10.1016/S0304-3975(00)00157-2.
  24. Pavel Hrubes and Pavel Pudlák. Random formulas, monotone circuits, and interpolation. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 121-131. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.20.
  25. Trinh Huynh and Jakob Nordström. On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 233-248. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214000.
  26. 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), Paris, France, July 4-7, 1994, pages 220-228. IEEE Computer Society, 1994. URL: https://doi.org/10.1109/LICS.1994.316069.
  27. Dmitry Itsykson and Artur Riazanov. Proof complexity of natural formulas via communication arguments. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), 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.
  28. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Ann. Pure Appl. Log., 171(1), 2020. URL: https://doi.org/10.1016/J.APAL.2019.102722.
  29. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Discret. Math., 3(2):255-265, 1990. URL: https://doi.org/10.1137/0403021.
  30. Jan Krajícek. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symb. Log., 62(2):457-486, 1997. URL: https://doi.org/10.2307/2275541.
  31. László Lovász, Moni Naor, Ilan Newman, and Avi Wigderson. Search problems in the decision tree model. SIAM J. Discret. Math., 8(1):119-132, 1995. URL: https://doi.org/10.1137/S0895480192233867.
  32. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with sunflowers. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 104:1-104:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.104.
  33. Marc Mezard, Giorgio Parisi, and Riccardo Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science (New York, N.Y.), 297:812-815, September 2002. URL: https://doi.org/10.1126/science.1073287.
  34. Toniann Pitassi and Robert Robere. Lifting nullstellensatz to monotone span programs over any field. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1207-1219. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188914.
  35. 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.
  36. Alexander A. Razborov. Applications of matrix methods to the theory of lower bounds in computational complexity. Comb., 10(1):81-93, 1990. URL: https://doi.org/10.1007/BF02122698.
  37. Robert Robere, Toniann Pitassi, Benjamin Rossman, and Stephen A. Cook. Exponential lower bounds for monotone span programs. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 406-415. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.51.
  38. Anastasia Sofronova and Dmitry Sokolov. A lower bound for k-dnf resolution on random CNF formulas via expansion. Electron. Colloquium Comput. Complex., TR22-054, 2022. URL: https://eccc.weizmann.ac.il/report/2022/054.
  39. Dmitry Sokolov. (semi)algebraic proofs over ±1 variables. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 78-90. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384288.
  40. Dmitry Sokolov. Random (log n)-CNF Are Hard for Cutting Planes (Again). In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, 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.
  41. Shuo Wang, Guangxu Yang, and Jiapeng Zhang. Communication Complexity of Set-Intersection Problems and Its Applications. Technical report, ECCC, 2023. URL: https://eccc.weizmann.ac.il/report/2023/164.
  42. 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, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649607.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact