TR19-170 Authors: Prerona Chatterjee, Mrinal Kumar, Adrian She, Ben Lee Volk

Publication: 27th November 2019 12:09

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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 by Kumar [Kum19], which showed a quadratic lower bound for $\text{homogeneous}$ 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 $\sum_{i = 1}^n x_i^n + \varepsilon(\mathbf{x})$, for a structured ``error polynomial'' $\varepsilon(\mathbf{x})$. To complete the proof, we then observe that the lower bound in [Kum19] is robust enough and continues to hold for all polynomials $\sum_{i = 1}^n x_i^n + \varepsilon(\mathbf{x})$, where $\varepsilon(\mathbf{x})$ has the appropriate structure.