Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

Abstract

We give an $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious arithmetic branching programs (ROABPs). The best time complexity known for blackbox polynomial identity testing (PIT) for this class was $n^{O(\log^2 n)}$ due to Forbes, Saptharishi, and Shpilka [Proceedings of the 2014 ACM Symposium on Theory of Computing, 2014, pp. 867--875]. Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the time complexity for the unknown-order ROABP with the known-order ROABP (due to Forbes and Shpilka [Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013, pp. 243--252]) and also with the depth-3 set-multilinear circuits (due to Agrawal, Saha, and Saxena [Proceedings of the 2013 ACM Symposium on Theory of Computing, 2013, pp. 321--330]). Our proof is simpler and involves a new technique called basis isolation. The depth-3 ...