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Hitting Sets for Regular Branching Programs

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

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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

Funding

  • Bogdanov, Andrej: Supported by RGC GRF CUHK 14209419 and CUHK 14209920. Part of the work was completed at the Simons Institute for the Theory of Computing, UC Berkeley.
  • Hoza, William M.: This work was done while the author was visiting the Simons Institute for the Theory of Computing.
  • Pyne, Edward: Supported by NSF grant CCF-1763299.

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

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