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 variable order. The best hitting-set known for this case had cost ( n w ) 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 complexity for the known-order case to n O ( log w ) . In particular, this gives the first polynomial time hitting-set for constant-width ROABP (known-order). However, our hitting-set works only over those fields whose characteristic is zero or large enough. To construct the hitting-set, we use the concept of the rank of partial derivative matrix. Unlike previous approaches whose basic building block is a monomial map, we use a polynomial map.The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost d O ( log w ) ( n w ) O ( log log w ) , where d is the individual degree. We improve this hitting-set complexity to ( nd w ) O ( log log w ) . We get this by achieving rank concentration more efficiently.
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