Blackbox identity testing for sum of special ROABPs and its border class

Pranav Bisht, Nitin Saxena

Abstract: We look at the problem of blackbox polynomial identity testing (PIT) for the model of read-once oblivious algebraic branching programs (ROABP), where the number of variables is logarithmic to the input size of ROABP. We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. This model is nontrivial due to the arbitrary individual-degree. We give the first poly(\(s\))-time blackbox PIT for sum of constant-many, size-\(s\), \(O(log s)\)-variate constant-width ROABPs. The previous best for this model was quasi-polynomial time (Gurjar et al, CCC'15; Computational Complexity'17) which is comparable to brute-force in the log-variate setting. We also show that we can work with unbounded-many such ROABPs if each ROABP computes a homogeneous polynomial (or more generally for degree-preserving sums). We also give poly-time PIT for the border. We introduce two new techniques, both of which also work for the border version of the stated models. (1) The leading-degree-part of an ROABP can be made syntactically homogeneous in the same width. (2) There is a direct reduction from PIT of sum-of-ROABPs to PIT of single ROABP (over any field). Our methods improve the time complexity for PIT of sum-of-ROABPs in the log-variate regime.

External IDs:dblp:journals/cc/BishtS21