VC Density of Set Systems Definable in Tree-Like Graphs

Authors Adam Paszke, Michał Pilipczuk

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Adam Paszke
  • University of Warsaw, Poland
Michał Pilipczuk
  • University of Warsaw, Poland

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Adam Paszke and Michał Pilipczuk. VC Density of Set Systems Definable in Tree-Like Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if phi(x,y) is a fixed CMSO_1 formula and C is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by phi in graphs from C have VC density at most |y|, which is the smallest bound that one could expect. We also show an analogous statement for the case when phi(x,y) is a CMSO_2 formula and C is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if C has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on C, the set systems definable by CMSO_1 (respectively, CMSO_2) formulas in graphs from C may have unbounded VC dimension, hence also unbounded VC density.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • treewidth
  • cliquewidth
  • definable sets
  • Vapnik-Chervonenkis density


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