Deterministic Identity Testing of Read-Once Algebraic Branching Programs

In this paper we study polynomial identity testing of sums of k read-once algebraic branching programs (Σk-RO-ABPs), generalizing the work of Shpilka and Volkovich [1, 2], who considered sums of k read-once formulas (Σk-RO-formulas). We show that Σk-RO-ABPs are strictly more powerful than Σk-RO-formulas, for any k ≤ ⌊n/2⌋, where n is the number of variables. Nevertheless, as a starting observation, we show that the generator given in [2] for testing a single RO-formula also works against a single RO-ABP. For the main technical part of this paper, we develop a property of polynomials called alignment. Using this property in conjunction with the hardness of representation approach of [1, 2], we obtain the following results for identity testing Σk-RO-ABPs, provided the underlying field has enough elements (more than kn suffices): 1. Given free access to the RO-ABPs in the sum, we get a deterministic algorithm that runs in time O(kns) + n, where s bounds the size of any largest RO-ABP given on the input. This implies we have a deterministic polynomial time algorithm for testing whether the sum of a constant number of RO-ABPs computes the zero polynomial. 2. Given black-box access to the RO-ABPs computing the individual polynomials in the sum, we get a deterministic algorithm that runs in time kn n) + n. 3. Finally, given only black-box access to the polynomial computed by the sum of the k RO-ABPs, we obtain an n n) time deterministic algorithm. Items 1. and 3. above strengthen two main results of [2] (Theorems 2 and 3, respectively, for the case of non-preprocessed Σk-RO-formulas).