Pseudorandom Generators for Width-3 Branching Programs

TL;DR

This paper presents new pseudorandom generators that efficiently fool width-3 ordered and unordered read-once branching programs, improving previous seed length bounds and employing novel analysis techniques.

Contribution

It introduces the first improved pseudorandom generators for width-3 ROBPs since 1992, using iterated milder restrictions and relabeling techniques.

Findings

Seed length for ordered ROBPs is $ ilde{O}( ext{log}(n) ext{log}(1/ extepsilon))$.

Seed length for unordered ROBPs is $ ilde{O}( ext{log}(n) ext{poly}(1/ extepsilon))$.

Achieves nearly optimal seed length for various classes of read-once polynomials and low-width ROBPs.

Abstract

We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/\epsilon))$ that $\epsilon$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/\epsilon))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work of Nisan [Combinatorica, 1992]. Our constructions are based on the `iterated milder restrictions' approach of Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered case, we combine iterated milder restrictions with the generator of Chattopadhyay et al. [CCC, 2018]. Two conceptual ideas that play an important role in our analysis are: (1) A relabeling technique allowing us to analyze a relabeled version of the given branching program, which turns out to be much easier. (2) Treating the number of colliding layers in a branching program as a progress measure and showing that it reduces significantly under pseudorandom restrictions. In addition, we achieve nearly optimal seed-length $\tilde{O}(\log(n/\epsilon))$ for the classes of: (1) read-once polynomials on $n$ variables, (2) locally-monotone ROBPs of length $n$ and width $3$ (generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length $n$ having a layer of width $2$ in every consecutive $\mathrm{poly}\log(n)$ layers.