Practical Computation of Graph VCDimension
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{VCdimension} of $H$ is the size of a largest shattered set in $V$. In this paper, we focus on the problem of computing the VCdimension of graphs. In particular, given a graph $G=(V,E)$, the VCdimension of $G$ is defined as the VCdimension 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 VCdimension 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 VCdimension, up to 8 in our experiments. As a sideproduct, we present several new bounds relating the graph VCdimension to other classical graph theoretical notions. We also establish the $W[1]$hardness of the graph VCdimension problem by extending a previous result for arbitrary set systems.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.07588
 arXiv:
 arXiv:2405.07588
 Bibcode:
 2024arXiv240507588C
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 Symposium on Experimental Algorithms (SEA) 2024, Jul 2024, Vienne, Austria