The composition complexity of majority

TL;DR

This paper establishes a tight lower bound on the number of local functions needed to compute the majority function through composition, revealing a fundamental complexity barrier and advancing understanding of circuit and branching program lower bounds.

Contribution

It proves an optimal lower bound on the composition overhead for majority, improving previous bounds and providing new techniques for circuit complexity lower bounds.

Findings

Lower bound of m ≥ Ω((n/k) log k) on the number of functions in composition

Recovery of known lower bounds for bounded-width branching programs via new proof methods

Introduction of novel techniques involving information flow and bootstrapping for lower bounds

Abstract

We study the complexity of computing majority as a composition of local functions: \[ \text{Maj}_n = h(g_1,\ldots,g_m), \] where each $g_j :\{0,1\}^{n} \to \{0,1\}$ is an arbitrary function that queries only $k \ll n$ variables and $h : \{0,1\}^m \to \{0,1\}$ is an arbitrary combining function. We prove an optimal lower bound of \[ m \ge \Omega\left( \frac{n}{k} \log k \right) \] on the number of functions needed, which is a factor $\Omega(\log k)$ larger than the ideal $m = n/k$. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority. Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions $g_j$, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).