Depth-3 Arithmetic Circuits for Sn2(X) and Extensions of the Graham-Pollack Theorem

We consider the problem of computing the second elementary symmetric polynomial S2 n(X) ∆ = ∑ 1≤i<j≤n XiXj using depth-three arithmetic circuits of the form ∑r i=1 ∏si j=1 Lij(X), where each Lij is a linear form in X1, . . . ,Xn. We consider this problem over several fields and determine exactly the number of multiplication gates required. The lower bounds are proved for inhomogeneous circuits where the Lij’s are allowed to have constants; the upper bounds are proved in the homogeneous model. For reals and rationals, the number of multiplication gates required is exactly n− 1; in most other cases, it is ⌈ n 2 ⌉ . This problem is related to the Graham-Pollack theorem in algebraic graph theory. In particular, our results answer the following question of Babai and Frankl: what is the minimum number of complete bipartite graphs required to cover each edge of a complete graph an odd number of times? We show that for infinitely many n, the answer is ⌈ n 2 ⌉