Hitting sets for regular branching programs

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 [EQUATION] 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 [EQUATION] 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 (log 2 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.