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.

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].

Our main technical contributions 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.