Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates

TL;DR

This paper explores the threshold dimension of graphs, showing how it relates to various graph parameters and how it can be realized as the AND of majority gates, with bounds for different graph classes.

Contribution

It establishes tight bounds on the threshold dimension in terms of graph parameters and connects it to the realization of graphic Boolean functions as AND of majority gates.

Findings

Threshold dimension bounds in terms of treewidth, degree, degeneracy, and other parameters.

Analysis of threshold dimension for random graphs and high-girth graphs.

Connection between threshold dimension and realization of Boolean functions as AND of majority gates.

Abstract

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by $\dimth(G)$, is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is the characteristic vector of a clique in $G$. A Boolean function $f$ for which there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean function. It follows that for a graph $G$, $\dimth(G)$ is precisely the minimum number of \emph{majority} gates whose AND (or conjunction) realizes the graphic Boolean function $f_G$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth.