Quadratic Lower Bounds for Algebraic Branching Programs and Formulas

Abstract

We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum _{i = 1}^n x_i^n$ has at least $\Omega (n^2)$ vertices. This improves upon the lower bound of $\Omega (n\log n)$, which follows from the classical result of Strassen (1973a) and Baur & Strassen (1983), and extends the results in Kumar (2019), which showed a quadratic lower bound for homogeneous ABPs computing the same polynomial.

Our proof relies on a notion of depth reduction, which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial $\sum _{i=1}^n x_i^n$ can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial $\sum _{i = 1}^n x_i^n + \varepsilon ({\bf x})$, for a structured “error polynomial” $\varepsilon ({\bf x})$. To complete the proof, we then observe that the lower bound in Kumar (2019) is robust enough and continues to hold for all polynomials $\sum _{i = 1}^n x_i^n + \varepsilon ({\bf x})$, where $\varepsilon ({\bf x})$ has the appropriate structure.

We also use our ideas to show an $\Omega (n^2)$ lower bound of the size of algebraic formulas computing the elementary symmetric polynomial of degree 0.1n on n variables. This is a slight improvement upon the prior best known formula lower bound (proved for a different polynomial) of $\Omega (n^2/\log n)$Kalorkoti (1985); Nechiporuk (1966); Shpilka & Yehudayoff (2010). Interestingly, this lower bound is asymptotically better than $n^2/\log n$, the strongest lower bound that can be proved using previous methods. This lower bound also matches the upper bound, due to Ben-Or, who showed that elementary symmetric polynomials can be computed by algebraic formula (in fact depth-3 formula) of size $O(n^2)$. Prior to this work, Ben-Or’s construction was known to be optimal only for algebraic formulas of depth-3 (Shpilka & Wigderson 2001).