Separating multilinear branching programs and formulas
Zeev Dvir, Guillaume Malod, Sylvain Perifel, Amir Yehudayoff
Proceedings of the 44th symposium on Theory of Computing - STOC '12
This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of multilinear ABPs to that of multilinear arithmetic formulas, and prove a tight super-polynomial separation between the two models. Specifically, we describe an explicit
... n-variate polynomial F that is computed by a linear-size multilinear ABP but every multilinear formula computing F must be of size n Ω(log n) . 1 for both the permanent and the determinant must be of super-polynomial size. Later, in  , Raz showed that multilinear circuits are super-polynomially stronger than multilinear formulas (see  for a simpler proof). Exponential lower bounds for constant depth multilinear circuits, as well as strong separations based on circuit-depth, were proved in  . Super-linear lower bounds for the size of arithmetic circuits were proved in  . In this article, we further extend this line of work by proving a super-polynomial separation between multilinear algebraic branching programs (ABPs) and multilinear formulas. As multilinear circuits can efficiently simulate multilinear ABPs, in particular, we strengthen the mentioned results of [7, 10] . Before stating our results we take a moment to formally define and to briefly motivate the two models (for more details, see the survey  ). An algebraic branching program (ABP) is a directed acyclic graph with two special nodes in it: a start-node and an end-node. The edges of the ABP are labeled by either variables or field elements. Every directed path γ from the start-node to the end-node computes the monomial f γ which is the product of all labels on the path γ. The ABP computes the polynomial f = γ f γ , where the sum is over all paths γ from start-node to end-node. A formula is a rooted directed binary tree (the edges are directed toward the root). The leaves of the formula are labeled by either variables or field elements. The inner nodes which have in-degree two are labeled by either + or ×. A formula computes a polynomial in the obvious way. Both ABPs and formulas have natural restrictions to the multilinear world. An ABP is multilinear if on every directed path from start-node to end-node no variable appears more than once. A formula is multilinear if every sub-formula in it computes a multilinear polynomial. ABPs capture the computational power of iterated matrix product: For every ABP of size s, there are poly(s) many matrices A 1 , A 2 , . . . of dimensions poly(s) × poly(s) with entries that are either variables or field elements, so that the polynomial computed by the ABP is the (1, 1) entry in the matrix A 1 A 2 · · · . In the other direction, for every s matrices of dimensions s × s, there is a (multi-start-node and multi-end-node) ABP of size poly(s) computing the product of the matrices. In fact, ABPs also capture the computational power of the determinant: For every ABP of size s, there is a matrix A of dimension poly(s) with entries that are either variables or field elements, so that the determinant of A is the polynomial the ABP computes [13, 4] , and the determinant can be computed by a polynomial-size ABP [1, 11, 3] . However, the known polynomial-size ABPs for the determinant are not multilinear, so the lower bound of  does not yield our result (by current knowledge). Formulas, on the other hand, capture a computational model in which every sub-computation can be used only once (as the underlying computation graph is a tree). Since formulas can be parallelized to have depth which is logarithmic in their size, they also capture the parallel time it takes to perform the computation. It is known that ABPs can efficiently simulate formulas  . Similar ideas show that multilinear ABPs can efficiently simulate multilinear formulas. A natural question is thus whether the other direction holds as well. We show that in the multilinear world it does not (a similar separation is believed to hold for general algebraic computation). This is the first separation between branching programs and formulas we are aware of. Theorem 1.1. For every positive integer n, there is a multilinear polynomial F = F n in n variables with zero-one coefficients so that the following holds: (i) There is a uniform algorithm that, given n, runs in time O(n) and outputs a multilinear branching program computing F .