A Quadratic Lower Bound for Algebraic Branching Programs and Formulas
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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. We also use our ideas to show an Ω(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 Ω(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].
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