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

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(logn) where n is the number of variables and w is the width of the ROABP. Even for a constantwidth ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set size for the known-order case to nO(logw). 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. A conference version of this paper appeared in the Proceedings of the 31st Computational Complexity Conference, 2016 [16]. ∗Supported by DFG grant TH 472/4. †Supported by DST-SERB. ACM Classification: F.2.2 AMS Classification: 68Q25, 68W30