A Quadratic Lower Bound 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 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 BenOr, 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, BenOr's construction was known to be optimal only for algebraic formulas of depth$3$ [SW01].
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.11793
 Bibcode:
 2019arXiv191111793C
 Keywords:

 Computer Science  Computational Complexity