# Hitting Sets for Regular Branching Programs

By Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne

Read the paper: ECCC

## Abstract (for specialists)

We construct improved hitting set generators (HSGs) for ordered (read-once) regular branching programs in two parameter regimes. First, we construct an explicit \(\epsilon\)-HSG for *unbounded-width* regular branching programs with a single accept state with seed length
\[
\widetilde{O}(\log n \cdot \log(1/\epsilon)),
\]
where \(n\) is the length of the program. Second, we construct an explicit \(\epsilon\)-HSG for width-\(w\) length-\(n\) regular branching programs with seed length
\[
\widetilde{O}\left(\log n \cdot \left(\sqrt{\log(1/\epsilon)} + \log w \right) + \log(1/\epsilon)\right).
\]
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 \(O(\log(wn/\epsilon) \cdot \log n)\). For regular programs, the state-of-the-art PRG, by Braverman, Rao, Raz, and Yehudayoff (FOCS 2010, SICOMP 2014), has seed length \(\widetilde{O}(\log(w/\epsilon) \cdot \log n)\), which beats Nisan's seed length when \(\log(w/\epsilon) = o(\log n)\). Taken together, our two new constructions beat Nisan's seed length in all parameter regimes except when \(\log w\) and \(\log(1/\epsilon)\) are *both* \(\Omega(\log n)\) (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 \(\log w = \Theta(\log(1/\epsilon)) = \Theta(\log n)\) 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.

## Not-so-abstract (for curious outsiders)

⚠️ *This summary might gloss over some important details.*

A "hitting set" is a set of binary strings \(H\) with a guarantee along the following lines: Let \(P\) be any property of binary strings such that (a) many strings have property \(P\) and (b) it's easy to check whether a given string has property \(P\). Then there is at least one string in \(H\) with property \(P\). (What exactly does "easy" mean? That varies depending on context. The meaning of "easy" in the context of our paper is somewhat technical, but it's related to computing with low memory.) In the paper, we present a couple of hitting sets that are smaller than all previously discovered hitting sets in some cases.

Manuscript posted online in November 2021. An earlier version of the manuscript, with Edward Pyne as the sole author, was posted online in October 2021.