On Algebraic Branching Programs of Small Width

Abstract

In 1979, Valiant showed that the complexity class VP e of families with polynomially bounded formula size is contained in the class VP s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes, we study the topological closure VP e , i.e., the class of polynomials that can be approximated arbitrarily closely by polynomials in VP e . We describe VP e using the well-known continuant polynomial (in characteristic different from 2). Further understanding this polynomial seems to be a promising route to new formula size lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992, Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP e . As a natural continuation of this work, we prove that the class VPN can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.