License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.78
URN: urn:nbn:de:0030-drops-127473
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12747/
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Paszke, Adam ; Pilipczuk, Michał

VC Density of Set Systems Definable in Tree-Like Graphs

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LIPIcs-MFCS-2020-78.pdf (0.8 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{paszke_et_al:LIPIcs:2020:12747,
  author =	{Adam Paszke and Micha{\l} Pilipczuk},
  title =	{{VC Density of Set Systems Definable in Tree-Like Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{78:1--78:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12747},
  URN =		{urn:nbn:de:0030-drops-127473},
  doi =		{10.4230/LIPIcs.MFCS.2020.78},
  annote =	{Keywords: treewidth, cliquewidth, definable sets, Vapnik-Chervonenkis density}
}

Keywords: treewidth, cliquewidth, definable sets, Vapnik-Chervonenkis density
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020

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