Quadratic Lower Bounds for Algebraic Branching Programs and Formulas

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 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=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) , 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 = 1^n x_i^n + ε ( x) , where ε ( x) 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 0.1 n 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) Kalorkoti (1985) ; Nechiporuk (1966) ; Shpilka Yehudayoff (2010) . 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 ( Shpilka Wigderson 2001 ).