Bounds on the Size of Small Depth Circuits for Approximating Majority

TL;DR

This paper establishes tight bounds on the size of small-depth circuits needed to approximate the majority function, showing exponential size growth with respect to input size for fixed depth.

Contribution

It provides a matching upper bound for the size of depth d circuits approximating majority for all d ≥ 3, extending prior lower bounds.

Findings

Minimum size of depth d circuits is exp(Θ(n^{1/(2d-2)}))

Matching upper bounds are established for d ≥ 3

Results extend previous bounds for depth 2 circuits

Abstract

In this paper, we show that for every constant $0 < \epsilon < 1/2$ and for every constant $d \geq 2$, the minimum size of a depth $d$ Boolean circuit that $\epsilon$-approximates Majority function on $n$ variables is exp$(\Theta(n^{1/(2d-2)}))$. The lower bound for every $d \geq 2$ and the upper bound for $d=2$ have been previously shown by O'Donnell and Wimmer [ICALP'07], and the contribution of this paper is to give a matching upper bound for $d \geq 3$.