Practical Computation of Graph VC-Dimension

Abstract: For any set system $H=(V,R), \ R \subseteq 2^V$, a subset $S \subseteq V$ is called \emph{shattered} if every $S' \subseteq S$ results from the intersection of $S$ with some set in $\R$. The \emph{VC-dimension} of $H$ is the size of a largest shattered set in $V$. In this paper, we focus on the problem of computing the VC-dimension of graphs. In particular, given a graph $G=(V,E)$, the VC-dimension of $G$ is defined as the VC-dimension of $(V, \mathcal N)$, where $\mathcal N$ contains each subset of $V$ that can be obtained as the closed neighborhood of some vertex $v \in V$ in $G$. Our main contribution is an algorithm for computing the VC-dimension of any graph, whose effectiveness is shown through experiments on various types of practical graphs, including graphs with millions of vertices. A key aspect of its efficiency resides in the fact that practical graphs have small VC-dimension, up to 8 in our experiments. As a side-product, we present several new bounds relating the graph VC-dimension to other classical graph theoretical notions. We also establish the $W[1]$-hardness of the graph VC-dimension problem by extending a previous result for arbitrary set systems.