Local Enumeration and Majority Lower Bounds

Authors Mohit Gurumukhani , Ramamohan Paturi, Pavel Pudlák, Michael Saks, Navid Talebanfard



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Mohit Gurumukhani
  • Cornell University, Ithaca, NY, USA
Ramamohan Paturi
  • Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA
Pavel Pudlák
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
Michael Saks
  • Department of Mathematics, Rutgers University, Piscataway, NJ, USA
Navid Talebanfard
  • University of Sheffield, UK
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic

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Mohit Gurumukhani, Ramamohan Paturi, Pavel Pudlák, Michael Saks, and Navid Talebanfard. Local Enumeration and Majority Lower Bounds. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 17:1-17:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.17

Abstract

Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Depth 3 circuits
  • k-CNF satisfiability
  • Circuit lower bounds
  • Majority function

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References

  1. Kazuyuki Amano. Depth-three circuits for inner product and majority functions. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation, ISAAC 2023, December 3-6, 2023, Kyoto, Japan, volume 283 of LIPIcs, pages 7:1-7:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.7.
  2. Tobias Brüggemann and Walter Kern. An improved deterministic local search algorithm for 3-SAT. Theor. Comput. Sci., 329(1-3):303-313, 2004. URL: https://doi.org/10.1016/j.tcs.2004.08.002.
  3. Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining circuit lower bound proofs for meta-algorithms. Comput. Complex., 24(2):333-392, 2015. URL: https://doi.org/10.1007/s00037-015-0100-0.
  4. Evgeny Dantsin, Andreas Goerdt, Edward A. Hirsch, Ravi Kannan, Jon M. Kleinberg, Christos H. Papadimitriou, Prabhakar Raghavan, and Uwe Schöning. A deterministic (2-2/(k+1))ⁿ algorithm for k-SAT based on local search. Theor. Comput. Sci., 289(1):69-83, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00174-8.
  5. Fedor V. Fomin, Serge Gaspers, Daniel Lokshtanov, and Saket Saurabh. Exact algorithms via monotone local search. J. ACM, 66(2):8:1-8:23, 2019. URL: https://doi.org/10.1145/3284176.
  6. Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. A variant of the VC-dimension with applications to depth-3 circuits. 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 72:1-72:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.72.
  7. Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit depth reductions. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ITCS.2021.24.
  8. Thomas Dueholm Hansen, Haim Kaplan, Or Zamir, and Uri Zwick. Faster k-SAT algorithms using biased-PPSZ. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 578-589. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316359.
  9. Johan Håstad, Stasys Jukna, and Pavel Pudlák. Top-down lower bounds for depth-three circuits. Comput. Complex., 5(2):99-112, 1995. URL: https://doi.org/10.1007/BF01268140.
  10. Timon Hertli. Breaking the PPSZ barrier for unique 3-SAT. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 600-611. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_50.
  11. Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC⁰. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 961-972. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.77.
  12. Peter Keevash. Hypergraph Turán problems. In Surveys in combinatorics 2011, volume 392 of London Math. Soc. Lecture Note Ser., pages 83-139. Cambridge Univ. Press, Cambridge, 2011. Google Scholar
  13. Konstantin Kutzkov and Dominik Scheder. Using CSP to improve deterministic 3-sat. CoRR, abs/1007.1166, 2010. URL: https://arxiv.org/abs/1007.1166.
  14. Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan. The composition complexity of majority. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 19:1-19:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.19.
  15. Andrea Lincoln and Adam Yedidia. Faster random k-CNF satisfiability. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 78:1-78:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.78.
  16. Burkhard Monien and Ewald Speckenmeyer. Solving satisfiability in less than 2ⁿ steps. Discret. Appl. Math., 10(3):287-295, 1985. URL: https://doi.org/10.1016/0166-218X(85)90050-2.
  17. Robin A. Moser and Dominik Scheder. A full derandomization of schöning’s k-SAT algorithm. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 245-252. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993670.
  18. Christos H. Papadimitriou. On selecting a satisfying truth assignment (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 163-169. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185365.
  19. Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. J. ACM, 52(3):337-364, 2005. URL: https://doi.org/10.1145/1066100.1066101.
  20. Ramamohan Paturi, Pavel Pudlák, and Francis Zane. Satisfiability coding lemma. Chic. J. Theor. Comput. Sci., 1999, 1999. URL: http://cjtcs.cs.uchicago.edu/articles/1999/11/contents.html.
  21. Ramamohan Paturi, Michael E. Saks, and Francis Zane. Exponential lower bounds for depth three boolean circuits. Comput. Complex., 9(1):1-15, 2000. URL: https://doi.org/10.1007/PL00001598.
  22. Dominik Scheder. PPSZ is better than you think. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 205-216. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00028.
  23. Dominik Scheder and Navid Talebanfard. Super strong ETH is true for PPSZ with small resolution width. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 3:1-3:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.3.
  24. Uwe Schöning. A probabilistic algorithm for k-SAT based on limited local search and restart. Algorithmica, 32(4):615-623, 2002. URL: https://doi.org/10.1007/s00453-001-0094-7.
  25. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Mathematical foundations of computer science (Proc. Sixth Sympos., Tatranská Lomnica, 1977), pages 162-176. Lecture Notes in Comput. Sci., Vol. 53, 1977. Google Scholar
  26. Nikhil Vyas and R. Ryan Williams. On super strong ETH. In Mikolás Janota and Inês Lynce, editors, Theory and Applications of Satisfiability Testing - SAT 2019 - 22nd International Conference, SAT 2019, Lisbon, Portugal, July 9-12, 2019, Proceedings, volume 11628 of Lecture Notes in Computer Science, pages 406-423. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24258-9_28.
  27. Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput., 42(3):1218-1244, 2013. URL: https://doi.org/10.1137/10080703X.
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