License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.43
URN: urn:nbn:de:0030-drops-186961
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18696/
Go to the corresponding LIPIcs Volume Portal

Dreier, Jan ; Mock, Daniel ; Rossmanith, Peter

Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond

pdf-format:
LIPIcs-ESA-2023-43.pdf (0.8 MB)

Abstract

It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-r minors have constant density. More precisely, the formulas are ∃ x₁… x_k#y φ(x_1,…,x_k, y) > N, where φ is an FO-formula. If φ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-r clique minors is subpolynomial, is impossible unless FPT = W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({>}) but not FOC₁(?).
In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.

BibTeX - Entry

@InProceedings{dreier_et_al:LIPIcs.ESA.2023.43,
  author =	{Dreier, Jan and Mock, Daniel and Rossmanith, Peter},
  title =	{{Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{43:1--43:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18696},
  URN =		{urn:nbn:de:0030-drops-186961},
  doi =		{10.4230/LIPIcs.ESA.2023.43},
  annote =	{Keywords: nowhere dense, sparsity, counting logic, dominating set, FPT}
}

Keywords: nowhere dense, sparsity, counting logic, dominating set, FPT
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI