Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs

TL;DR

This paper presents improved polynomial-size hitting sets for specific classes of read-once oblivious arithmetic branching programs, notably achieving the first polynomial-size hitting set for constant-width, known-order ROABPs, and enhancing bounds for unordered cases.

Contribution

It introduces new hitting set constructions with reduced sizes for two special cases of ROABPs, including the first polynomial-size set for constant-width, known-order ROABPs.

Findings

Achieved polynomial-size hitting set for constant-width, known-order ROABPs.

Improved hitting set size for unordered width-w ROABPs to $(ndw)^{O( ext{log log w})}$.

Utilized the rank of the partial derivative matrix and polynomial mappings for construction.

Abstract

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 previously 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 $(ndw)^{O(\log \log w)}$.