On the Power of Regular and Permutation Branching Programs

We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation. - Regular SOBPs of length n and width ?w(n+1)/2? can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs. - There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width. - There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs. - Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs

Optimal Error Pseudodistributions for Read-Once Branching Programs

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has been no improvement on Nisan's construction for the case $n=w$, which is most relevant to space-bounded derandomization. Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length $\tilde{O}(\log n\cdot \log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length $O(\log n\cdot \log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting sets. In this work, we construct a PRPD with seed length $$O(\log n\cdot \log (nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$ This improves upon the construction in [BCG18] by a $O(\log\log(1/\varepsilon))$ factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg

Singular Value Approximation and Sparsifying Random Walks on Directed Graphs

In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs (Spielman Teng STOC 2004), standard approximation for directed graphs (Cohen et. al. STOC 2017), and unit-circle approximation for directed graphs (Ahmadinejad et. al. FOCS 2020). Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of random-walk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as $\ell$-step random walks on such graphs, for any $\ell\leq \text{poly}(n)$. Combined with the Eulerian scaling algorithms of (Cohen et. al. FOCS 2018), given an arbitrary (not necessarily Eulerian) directed graph and a set $S$ of vertices, we can approximate the stationary probability mass of the $(S,S^c)$ cut in an $\ell$-step random walk to within a multiplicative error of $1/\text{polylog}(n)$ and an additive error of $1/\text{poly}(n)$ in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.Comment: FOCS 202

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum