eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
3:1
3:22
10.4230/LIPIcs.CCC.2022.3
article
Hitting Sets for Regular Branching Programs
Bogdanov, Andrej
1
Hoza, William M.
2
https://orcid.org/0000-0001-5162-9181
Prakriya, Gautam
1
Pyne, Edward
3
The Chinese University of Hong Kong, China
Simons Institute for the Theory of Computing, Berkeley, CA, USA
Harvard University, Cambridge, MA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.3/LIPIcs.CCC.2022.3.pdf
Pseudorandomness
hitting set generators
space-bounded computation