A quadratic lower bound for homogeneous algebraic branching programs

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in $$\mathbb{F}[x_1, x_2, \ldots , x_n]$$ . An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous if the polynomial computed at every vertex is homogeneous. In this paper, we show that any homogeneous algebraic branching program which computes the polynomial $$x^n_1 + x^n_2 + \cdots + x^n_n$$ has at least $$\Omega(n^2)$$ vertices (and hence edges). To the best of our knowledge, this seems to be the first non-trivial super-linear lower bound on the number of vertices for a general homogeneous ABP and slightly improves the known lower bound of $$\Omega(n \,{\rm log}\, n)$$ on the number of edges in a general (possibly non-homogeneous) ABP, which follows from the classical results of Strassen (Numer Math 20:238–251, 1973) and Baur and Strassen (Theor Comput Sci 22:317–330, 1983). On the way, we also get an alternate and unified proof of an $$\Omega(n \,{\rm log}\, n)$$ lower bound on the size of a homogeneous arithmetic circuit (follows from the work of Strassen (1973) and Baur & Strassen (1983)), and an n/2 lower bound $$(n \,{\rm over}\, \mathbb{R})$$ on the determinantal complexity of an explicit polynomial (Mignon and Ressayre in Int Math Res Notes 2004(79):4241–4253, 2004; Cai et al. in Comput Complex 19(1):37–56, 2010, http://dx.doi.org/10.1007/s00037-009-0284-2 ; Yabe in CoRR, 2015, http://arxiv.org/abs/1504.00151 ). These are currently the best lower bounds known for these problems for any explicit polynomial and were originally proved nearly two decades apart using seemingly different proof techniques.