A Quadratic Lower Bound for Algebraic Branching Programs and Formulas

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 Baur and Strassen [Str73, BS83], and extends the results in [K19], which showed a quadratic lower bound for \emph{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 $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$, for a structured "error polynomial" $\epsilon(x_1, \ldots, x_n)$. To complete the proof, we then observe that the lower bound in [K19] is robust enough and continues to hold for all polynomials $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$, where $\epsilon(x_1, \ldots, x_n)$ 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 $\Omega(n^2/\log n)$ [Nec66, K85, SY10]. 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-$3$ [SW01].