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#SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank

Authors Nutan Limaye , Adarsh Srinivasan , Srikanth Srinivasan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • probabilistic polynomials
  • probabilistic rank
  • circuit satisfiability
  • circuit lower bounds
  • polynomial method
  • threshold circuits

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Abstract

There is a large body of work that shows how to leverage lower bound techniques for circuit classes to obtain satisfiability algorithms that run in better than brute-force time [Ramamohan Paturi et al., 1997; Ryan Williams, 2014]. For circuits with threshold gates, there are several such algorithms based on either  
- Probabilistic Representations by low-degree polynomials, which allow for the use of fast polynomial evaluation algorithms, or 
- Low rank, which allows for an efficient reduction to rectangular matrix multiplication.  In this paper, we use a related notion of probabilistic rank to obtain satisfiability algorithms for circuit classes contained in ACC⁰∘3-PTF, i.e. constant-depth circuits with modular counting gates and a single layer of degree-3 polynomial threshold functions. 
Even for the special case of a single 3-PTF, it is not clear how to use either of the above two strategies to get a non-trivial satisfiability algorithm. The best known algorithm in this case previously was based on memoization and yields worse guarantees than our algorithm.

Cite As Get BibTex

Nutan Limaye, Adarsh Srinivasan, and Srikanth Srinivasan. #SAT-Algorithms for Classes of Threshold Circuits Based on Probabilistic Rank. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.67

Author Details

Nutan Limaye
  • {IT} {University} of {Copenhagen}, Denmark
Adarsh Srinivasan
  • Rutgers University, Piscataway, NJ, USA
Srikanth Srinivasan
  • {University} of {Copenhagen}, Denmark

Funding

  • Limaye, Nutan: Received funding from the Independent Research Fund Denmark (grant agreement No. 10.46540/3103-00116B) and is also supported by the Basic Algorithms Research Copenhagen (BARC), funded by VILLUM Foundation Grants 16582 and 54451, and Digital Research Centre Denmark, project P40.
  • Srinivasan, Adarsh: Supported by the National Science Foundation under Grants CCF-2313372 and CCF-2443697. Part of this work was done during a visit to ITU Copenhagen and BARC funded by Basic Algorithms Research Copenhagen(BARC), supported by VILLUM Foundation Grants 16582 and 54451, and while at INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria. This work was partially funded from the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).
  • Srinivasan, Srikanth: Funded by the European Research Council (ERC) under grant agreement no. 101125652 (ALBA).

References

  1. Eric Allender and Vivek Gore. On strong separations from AC⁰. In International Symposium on Fundamentals of Computation Theory, pages 1-15. Springer, 1991. URL: https://doi.org/10.1007/3-540-54458-5_44.
  2. Eric Allender and Vivek Gore. A uniform circuit lower bound for the permanent. SIAM J. Comput., 23(5):1026-1049, 1994. URL: https://doi.org/10.1137/S0097539792233907.
  3. Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. 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 467-476. IEEE, IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.57.
  4. Josh Alman and Kevin Rao. Faster walsh-hadamard and discrete fourier transforms from matrix non-rigidity. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 455-462. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585188.
  5. Josh Alman and R. Ryan Williams. Probabilistic rank and matrix rigidity. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 641-652. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055484.
  6. Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 136-150. IEEE, IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.18.
  7. Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #sat algorithm for small constant-depth circuits with PTF gates. Algorithmica, 84(4):1132-1162, 2022. URL: https://doi.org/10.1007/s00453-021-00915-7.
  8. Richard Beigel and Jun Tarui. On ACC. Comput. Complex., 4:350-366, 1994. URL: https://doi.org/10.1007/BF01263423.
  9. Timothy M. Chan. Orthogonal range searching in moderate dimensions: k-d trees and range trees strike back. In Boris Aronov and Matthew J. Katz, editors, 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 27:1-27:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.27.
  10. Ashok K. Chandra, Larry J. Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM J. Comput., 13(2):423-439, 1984. URL: https://doi.org/10.1137/0213028.
  11. Lijie Chen and R. Ryan Williams. Stronger connections between circuit analysis and circuit lower bounds, via pcps of proximity. In Amir Shpilka, editor, 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, volume 137 of LIPIcs, pages 19:1-19:43. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.19.
  12. R Chen, R Santhanam, and S Srinivasan. Average-case lower bounds and satisfiability algorithms for small threshold circuits. Theory of Computing, 14, 2018. URL: https://doi.org/10.4086/toc.2018.v014a009.
  13. Chao-Kong Chow. On the characterization of threshold functions. In 2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961), pages 34-38. IEEE, 1961. URL: https://doi.org/10.1109/FOCS.1961.24.
  14. Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM J. Comput., 11(3):467-471, 1982. URL: https://doi.org/10.1137/0211037.
  15. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL: https://doi.org/10.1145/12130.12132.
  16. Johan Håstad. Computational limitations for small depth circuits. PhD thesis, Massachusetts Institute of Technology, 1986. Google Scholar
  17. Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for ac^0. 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, SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.77.
  18. Valentine Kabanets and Zhenjian Lu. Satisfiability and derandomization for small polynomial threshold circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.46.
  19. Alexis Maciel and Denis Thérien. Threshold circuits of small majority-depth. Inf. Comput., 146(1):55-83, 1998. URL: https://doi.org/10.1006/inco.1998.2732.
  20. Saburo Muroga. Threshold logic and its applications. John Wiley & Sons, 1971. Google Scholar
  21. Cody Murray and R. Ryan Williams. Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for NP and NQP. 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 890-901. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188910.
  22. Eduard Ivanovich Nechiporuk. On a boolean function. In Dokl. Akad. Nauk SSSR, volume 169(4), pages 765-766. Russian Academy of Sciences, 1966. Google Scholar
  23. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.
  24. Ramamohan Paturi, Pavel Pudlák, and Francis Zane. Satisfiability coding lemma. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 566-574. IEEE, IEEE Computer Society, 1997. URL: https://doi.org/10.1109/SFCS.1997.646146.
  25. Ninad Rajgopal, Rahul Santhanam, and Srikanth Srinivasan. Deterministically counting satisfying assignments for constant-depth circuits with parity gates, with implications for lower bounds. In Igor Potapov, Paul G. Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, volume 117 of LIPIcs, pages 78:1-78:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.78.
  26. Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mat. Zametki, 41(4):598-607, 1987. Google Scholar
  27. Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, and Junichi Teruyama. Improved exact algorithms for mildly sparse instances of max SAT. Theor. Comput. Sci., 697:58-68, 2017. URL: https://doi.org/10.1016/j.tcs.2017.07.011.
  28. Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, and Junichi Teruyama. Bounded depth circuits with weighted symmetric gates: Satisfiability, lower bounds and compression. J. Comput. Syst. Sci., 105:87-103, 2019. URL: https://doi.org/10.1016/j.jcss.2019.04.004.
  29. Rahul Santhanam. Fighting perebor: New and improved algorithms for formula and QBF satisfiability. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 183-192. IEEE, IEEE Computer Society, 2010. URL: https://doi.org/10.1109/FOCS.2010.25.
  30. Kazuhisa Seto and Suguru Tamaki. A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Comput. Complex., 22(2):245-274, 2013. URL: https://doi.org/10.1007/s00037-013-0067-7.
  31. Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. SIAM J. Comput., 42(6):2329-2374, 2013. URL: https://doi.org/10.1137/100785260.
  32. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL: https://doi.org/10.1145/28395.28404.
  33. Heribert Vollmer. Introduction to circuit complexity: a uniform approach. Springer Science & Business Media, 1999. URL: https://doi.org/10.1007/978-3-662-03927-4.
  34. R. Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. Theory Comput., 14(1):1-25, 2018. URL: https://doi.org/10.4086/toc.2018.v014a017.
  35. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.023.
  36. Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput., 42(3):1218-1244, 2013. URL: https://doi.org/10.1137/10080703X.
  37. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 664-673. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591811.
  38. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2:1-2:32, 2014. URL: https://doi.org/10.1145/2559903.
  39. Andrew Chi-Chih Yao. On ACC and threshold circuits. In 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, October 22-24, 1990, Volume II, pages 619-627. IEEE, IEEE Computer Society, 1990. URL: https://doi.org/10.1109/FSCS.1990.89583.
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