Depth-3 Arithmetic Circuits for S^2_n(X) and Extensions of the Graham-Pollack Theorem

TL;DR

This paper determines the exact number of multiplication gates needed to compute the second elementary symmetric polynomial using depth-three circuits over various fields, connecting algebraic circuit complexity with a graph covering problem.

Contribution

It provides precise bounds for the size of depth-three arithmetic circuits computing S^2_n(X) over different fields, extending the Graham-Pollack theorem to algebraic circuit complexity.

Findings

Exact bounds for multiplication gates over reals and rationals: n-1.

Bounds over other fields: ceiling of n/2.

Connection established between circuit complexity and bipartite graph coverings.

Abstract

We consider the problem of computing the second elementary symmetric polynomial S^2_n(X) using depth-three arithmetic circuits of the form "sum of products of linear forms". 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 linear forms 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 \ceil{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 \ceil{n/2}.