Deterministic Black-Box Identity Testing $pi$-Ordered Algebraic Branching Programs

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source s and sink t, whose edges are labeled by variables taken from the set {x1, x2, . . . , xn} or field constants. It computes the sum of weights of all paths from s to t, where the weight of a path is defined as the product of edge-labels on the path. Given a permutation π of the n variables, for a π-ordered ABP (π-OABP), for any directed path p from s to t, a variable can appear at most once on p, and the order in which variables appear on p must respect π. An ABP A is said to be of read r, if any variable appears at most r times in A. Our main result pertains to the identity testing problem, i.e. the problem of deciding whether a given n-variate polynomial is identical to the zero polynomial or not. Over any field F and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read r π-OABP computable polynomials can be tested in DTIME[2 log r·log 2 n log log ]. In case F is a finite field, the above time bound holds provided the identity testing algorithm is allowed to make queries to extension fields of F. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size Ω(2/n) and read Ω(2/n). We give a multilinear polynomial p in 2n+1 variables over some specifically selected field G, such that any OABP computing pmust read some variable at least 2 times. We prove a strict separation for the computational power of read (r−1) and read r OABPs. Namely, we show that the elementary symmetric polynomial of degree r in n variables can be computed by a size O(rn) read r OABP, but not by a read (r− 1) OABP, for any 0 < 2r − 1 ≤ n. Finally, we give an example of a polynomial p and two variables orders π 6= π′, such that p can be computed by a read-once π-OABP, but where any π′-OABP computing p must read some variable at least 2 times.