A Quadratic Lower Bound for Algebraic Branching Programs
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We show that any Algebraic Branching Program (ABP) computing the polynomial ∑_i = 1^n x_i^n has at least Ω(n^2) vertices. This improves upon the lower bound of Ω(nlog 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 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=1^n x_i^n can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial ∑_i = 1^n x_i^n + ϵ(x_1, ..., x_n), for a structured "error polynomial" ϵ(x_1, ..., 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 ∑_i = 1^n x_i^n + ϵ(x_1, ..., x_n), where ϵ(x_1, ..., x_n) has the appropriate structure.
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