The isomorphism problem for read-once branching programs and arithmetic circuits

We investigate the computational complexity of the isomorphism problem for read-once branching programs (1-BPI): upon input of two read-once branching programs B-0 and B-1, decide whether there exists a permutation of the variables of B-1 such that it becomes equivalent to B-0. Our main result is that 1-BPI cannot be NP-hard unless the polynomial hierarchy collapses. The result is extended to the isomorphism problem for arithmetic circuits over large enough fields. We use the known arithmetization of read-once branching programs and arithmetic circuits into multivariate polynomials over the rationals. Hence, another way of stating our result is: the isomorphism problem for multivariate polynomials over large enough fields is not NP-hard unless the polynomial hierarchy collapses. We derive this result by providing a two-round interactive proof for the nonisomorphism problem for multivariate polynomials. The protocol is based on the Schwartz-Zippel theorem for probabilistically checking polynomial identities. Finally, we show that there is a perfect zero-knowledge interactive proof for the isomorphism problem for multivariate polynomials.