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Quadratic Lower Bounds for Algebraic Branching Programs and Formulas
Journal
Computational Complexity
ISSN
10163328
Date Issued
2022-12-01
Author(s)
Chatterjee, Prerona
Kumar, Mrinal
She, Adrian
Lee Volk, Ben
Abstract
We show that any Algebraic Branching Program (ABP) computing the polynomial ∑i=1nxin has at least Ω (n2) vertices. This improves upon the lower bound of Ω (nlog 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 ∑i=1nxin can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑i=1nxin+ε(x), for a structured “error polynomial” ε(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 ∑i=1nxin+ε(x), where ε(x) has the appropriate structure. We also use our ideas to show an Ω (n2) 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 Ω (n2/ log n) Kalorkoti (1985); Nechiporuk (1966); Shpilka & Yehudayoff (2010). Interestingly, this lower bound is asymptotically better than n2/ 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(n2). Prior to this work, Ben-Or’s construction was known to be optimal only for algebraic formulas of depth-3 (Shpilka & Wigderson 2001).
Volume
31
Subjects