Volume 13 (2017) Article 2 pp. 1-21
CCC 2016 Special Issue
Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs
by
Revised: April 9, 2017
Published: May 15, 2017
We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known order of the variables. The best previously known hitting-set for this case had size $(nw)^{O(\log n)}$ where $n$ is the number of variables and $w$ is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set size for the known-order case to $n^{O(\log w)}$. In particular, this gives the first polynomial-size hitting-set for constant-width ROABP (known-order). However, our hitting-set only works when the characteristic of the field is zero or large enough. To construct the hitting-set, we use the concept of the rank of the partial derivative matrix. Unlike previous approaches which build up from mapping variables to monomials, we map variables to polynomials directly.
The second case we consider is that of polynomials computable by width-$w$ ROABPs in any order of the variables. The best known hitting-set for this case had size $d^{O(\log w)}(nw)^{O(\log \log w)}$, where $d$ is the individual degree. We improve the hitting-set size to $(ndw)^{O(\log \log w)}$.