Abstract
We construct improved hitting set generators (HSGs) for ordered (readonce) regular branching programs in two parameter regimes. First, we construct an explicit εHSG for unboundedwidth 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 εHSG for widthw lengthn 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 nonregular) branching programs with seed length O(log(wn/ε) ⋅ log n). For regular programs, the stateoftheart 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.
BibTeX  Entry
@InProceedings{bogdanov_et_al:LIPIcs.CCC.2022.3,
author = {Bogdanov, Andrej and Hoza, William M. and Prakriya, Gautam and Pyne, Edward},
title = {{Hitting Sets for Regular Branching Programs}},
booktitle = {37th Computational Complexity Conference (CCC 2022)},
pages = {3:13:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772419},
ISSN = {18688969},
year = {2022},
volume = {234},
editor = {Lovett, Shachar},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16565},
URN = {urn:nbn:de:0030drops165658},
doi = {10.4230/LIPIcs.CCC.2022.3},
annote = {Keywords: Pseudorandomness, hitting set generators, spacebounded computation}
}
Keywords: 

Pseudorandomness, hitting set generators, spacebounded computation 
Collection: 

37th Computational Complexity Conference (CCC 2022) 
Issue Date: 

2022 
Date of publication: 

11.07.2022 