Document Open Access Logo

Hitting Sets for Regular Branching Programs

Authors Andrej Bogdanov, William M. Hoza , Gautam Prakriya, Edward Pyne



PDF
Thumbnail PDF

File

LIPIcs.CCC.2022.3.pdf
  • Filesize: 0.85 MB
  • 22 pages

Document Identifiers

Author Details

Andrej Bogdanov
  • The Chinese University of Hong Kong, China
William M. Hoza
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
Gautam Prakriya
  • The Chinese University of Hong Kong, China
Edward Pyne
  • Harvard University, Cambridge, MA, USA

Acknowledgements

We thank Sumegha Garg and Salil Vadhan for many helpful discussions, and Salil Vadhan and Chin Ho Lee for comments on a draft of this paper.

Cite AsGet BibTex

Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting Sets for Regular Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 3:1-3:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.3

Abstract

We construct improved hitting set generators (HSGs) for ordered (read-once) regular branching programs in two parameter regimes. First, we construct an explicit ε-HSG for unbounded-width regular branching programs with a single accept state with seed length Õ(log n ⋅ log(1/ε)), where n is the length of the program. Second, we construct an explicit ε-HSG for width-w length-n regular branching programs with seed length Õ(log n ⋅ (√{log(1/ε)} + log w) + log(1/ε)). For context, the "baseline" in this area is the pseudorandom generator (PRG) by Nisan (Combinatorica 1992), which fools ordered (possibly non-regular) branching programs with seed length O(log(wn/ε) ⋅ log n). For regular programs, the state-of-the-art PRG, by Braverman, Rao, Raz, and Yehudayoff (FOCS 2010, SICOMP 2014), has seed length Õ(log(w/ε) ⋅ log n), which beats Nisan’s seed length when log(w/ε) = o(log n). Taken together, our two new constructions beat Nisan’s seed length in all parameter regimes except when log w and log(1/ε) are both Ω(log n) (for the construction of HSGs for regular branching programs with a single accept vertex). Extending work by Reingold, Trevisan, and Vadhan (STOC 2006), we furthermore show that an explicit HSG for regular branching programs with a single accept vertex with seed length o(log² n) in the regime log w = Θ(log(1/ε)) = Θ(log n) would imply improved HSGs for general ordered branching programs, which would be a major breakthrough in derandomization. Pyne and Vadhan (CCC 2021) recently obtained such parameters for the special case of permutation branching programs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandomness
  • hitting set generators
  • space-bounded computation

Metrics

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

References

  1. AmirMahdi Ahmadinejad, Jonathan A. Kelner, Jack Murtagh, John Peebles, Aaron Sidford, and Salil P. Vadhan. High-precision estimation of random walks in small space. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1295-1306, 2020. Google Scholar
  2. Miklós Ajtai, János Komlós, and E. Szemerédi. Deterministic simulation in LOGSPACE. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 132-140, 1987. Google Scholar
  3. Mihir Bellare, Oded Goldreich, and Shafi Goldwasser. Randomness in interactive proofs. Computational Complexity, 3(4):319-354, 1993. Google Scholar
  4. Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting sets for regular branching programs. Technical Report TR21-143, Electronic Colloquium on Computational Complexity (ECCC), 2021. URL: https://eccc.weizmann.ac.il/report/2021/143/.
  5. Mark Braverman, Gil Cohen, and Sumegha Garg. Pseudorandom pseudo-distributions with near-optimal error for read-once branching programs. SIAM J. Comput., 49(5), 2020. URL: https://doi.org/10.1137/18M1197734.
  6. Mark Braverman, Anup Rao, Ran Raz, and Amir Yehudayoff. Pseudorandom generators for regular branching programs. SIAM J. Comput., 43(3):973-986, 2014. URL: https://doi.org/10.1137/120875673.
  7. Joshua Brody and Elad Verbin. The coin problem and pseudorandomness for branching programs. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 30-39, 2010. URL: https://doi.org/10.1109/FOCS.2010.10.
  8. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. Theory Comput., 15:1-26, 2019. URL: https://doi.org/10.4086/toc.2019.v015a010.
  9. Eshan Chattopadhyay and Jyun-Jie Liao. Optimal error pseudodistributions for read-once branching programs. In Proceedings of the 35th Computational Complexity Conference (CCC), pages 25:1-25:27, 2020. Google Scholar
  10. Kuan Cheng and William M. Hoza. Hitting sets give two-sided derandomization of small space. In Proceedings of the 35th Computational Complexity Conference (CCC), pages 10:1-10:25, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.10.
  11. Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma. Error Reduction for Weighted PRGs Against Read Once Branching Programs. In Proceedings of the 36th Computational Complexity Conference (CCC), pages 22:1-22:17, 2021. Google Scholar
  12. Michael B. Cohen, Jonathan Kelner, Rasmus Kyng, John Peebles, Richard Peng, Anup B. Rao, and Aaron Sidford. Solving directed Laplacian systems in nearly-linear time through sparse LU factorizations. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 898-909, 2018. Google Scholar
  13. Anindya De. Pseudorandomness for permutation and regular branching programs. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity (CCC), pages 221-231, 2011. URL: https://doi.org/10.1109/CCC.2011.23.
  14. Dean Doron, Pooya Hatami, and William M. Hoza. Log-Seed Pseudorandom Generators via Iterated Restrictions. In Proceedings of the 35th Annual Computational Complexity Conference (CCC), pages 6:1-6:36, 2020. Google Scholar
  15. Michael A. Forbes and Zander Kelley. Pseudorandom generators for read-once branching programs, in any order. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 946-955, 2018. URL: https://doi.org/10.1109/FOCS.2018.00093.
  16. Oded Goldreich. A sample of samplers: A computational perspective on sampling. In Studies in Complexity and Cryptography: Miscellanea on the Interplay between Randomness and Computation, volume 6650 of Lecture Notes in Computer Science, pages 302-332. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_24.
  17. William M. Hoza. Better pseudodistributions and derandomization for space-bounded computation. In Proceedings of the 25th International Conference on Randomization and Computation (RANDOM), pages 28:1-28:23, 2021. Google Scholar
  18. William M. Hoza, Edward Pyne, and Salil P. Vadhan. Pseudorandom generators for unbounded-width permutation branching programs. In James R. Lee, editor, Proceedings of the 12th Innovations in Theoretical Computer Science Conference (ITCS), pages 7:1-7:20, 2021. Google Scholar
  19. William M. Hoza and David Zuckerman. Simple optimal hitting sets for small-success RL. SIAM J. Comput., 49(4):811-820, 2020. URL: https://doi.org/10.1137/19M1268707.
  20. Michal Koucký, Prajakta Nimbhorkar, and Pavel Pudlák. Pseudorandom generators for group products: extended abstract. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 263-272, 2011. URL: https://doi.org/10.1145/1993636.1993672.
  21. Chin Ho Lee. Fourier Bounds and Pseudorandom Generators for Product Tests. In Proceedings of the 34th Annual Computational Complexity Conference (CCC), pages 7:1-7:25, 2019. Google Scholar
  22. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. SIAM J. Comput., 42(3):1275-1301, 2013. Google Scholar
  23. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. Google Scholar
  24. Edward Pyne and Salil Vadhan. Pseudodistributions That Beat All Pseudorandom Generators (Extended Abstract). In Proceedings of the 36th Annual Computational Complexity Conference (CCC), pages 33:1-33:15, 2021. Google Scholar
  25. Edward Pyne and Salil Vadhan. Deterministic approximation of random walks via queries in graphs of unbounded size. In Proceedings of the 5th Symposium on Simplicity in Algorithms (SOSA), pages 57-67, 2022. Google Scholar
  26. Edward Pyne and Salil P. Vadhan. Limitations of the Impagliazzo-Nisan-Wigderson pseudorandom generator against permutation branching programs. In Proceedings of the 27th International Computing and Combinatorics Conference (COCOON), pages 3-12, 2021. URL: https://doi.org/10.1007/978-3-030-89543-3_1.
  27. Omer Reingold, Thomas Steinke, and Salil Vadhan. Pseudorandomness for regular branching programs via Fourier analysis. In Proceedings of the 17th International Workshop on Randomization and Computation (RANDOM), pages 655-670, 2013. Google Scholar
  28. Omer Reingold, Luca Trevisan, and Salil Vadhan. Pseudorandom walks in regular digraphs and the RL vs. L problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 457-466, 2006. Google Scholar
  29. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics, 155(1), January 2002. Google Scholar
  30. Eyal Rozenman and Salil Vadhan. Derandomized squaring of graphs. In Proceedings of the 8th International Workshop on Randomization and Computation (RANDOM), pages 436-447, 2005. Google Scholar
  31. Thomas Steinke. Pseudorandomness for permutation branching programs without the group theory. Technical Report TR12-083, Electronic Colloquium on Computational Complexity (ECCC), July 2012. URL: http://eccc.hpi-web.de/report/2012/083/.
  32. R. Ryan Williams. Counting Solutions to Polynomial Systems via Reductions. In Proceedings of the 1st Symposium on Simplicity in Algorithms (SOSA), pages 6:1-6:15, Dagstuhl, Germany, 2018. Google Scholar
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