Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs

Gurjar, Rohit ; Korwar, Arpita ; Saxena, Nitin (2016) Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs In: 31st Conference on Computational Complexity (CCC 2016).

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Official URL: http://drops.dagstuhl.de/opus/volltexte/2016/5843

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 variable order. The best hitting-set known for this case had cost (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 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 starting point is a monomial map, we use a polynomial map directly. The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost d^{O(log(w))}(nw)^{O(log(log(w)))}, where d is the individual degree. We improve this hitting-set complexity to (ndw)^{O(log(log(w)))}. We get this by achieving rank concentration more efficiently.

Item Type:Conference or Workshop Item (Paper)
Source:Copyright of this article belongs to Schloss Dagstuhl--Leibniz-Zentrum für Informatik.
Keywords:PIT; Hitting-Set; Constant-Width ROABPs; Commutative ROABPs.
ID Code:122768
Deposited On:16 Aug 2021 06:48
Last Modified:16 Aug 2021 06:48

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