Practical Computation of Graph VC-Dimension

Authors David Coudert, Mónika Csikós, Guillaume Ducoffe, Laurent Viennot



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David Coudert
  • Université Côte d'Azur, Inria, I3S, CNRS, France
Mónika Csikós
  • IRIF, CNRS and Université Paris Cité, Paris, France
Guillaume Ducoffe
  • University of Bucharest, Romania
  • National Institute for Research and Development in Informatics, Bucharest, Romania
Laurent Viennot
  • Inria, DI ENS, Paris, France

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David Coudert, Mónika Csikós, Guillaume Ducoffe, and Laurent Viennot. Practical Computation of Graph VC-Dimension. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SEA.2024.8

Abstract

For any set system ℋ = (V,ℛ), ℛ ⊆ 2^V, a subset S ⊆ V is called shattered if every S' ⊆ S results from the intersection of S with some set in ℛ. The 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, N), where N contains each subset of V that can be obtained as the closed neighborhood of some vertex v ∈ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Algorithm design techniques
Keywords
  • VC-dimension
  • graph
  • algorithm

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