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.STACS.2023.46
URN: urn:nbn:de:0030-drops-176981
URL: https://drops.dagstuhl.de/opus/volltexte/2023/17698/
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Majewski, Konrad ; Pilipczuk, Michał ; Sokołowski, Marek

Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number

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LIPIcs-STACS-2023-46.pdf (1.0 MB)


Abstract

Let 𝜑 be a sentence of CMSO₂ (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether 𝜑 is satisfied in G. The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time O_{𝜑,k}(log n).
By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time O_k(log n). Our data structure also works in a larger generality of relational structures over binary signatures.

BibTeX - Entry

@InProceedings{majewski_et_al:LIPIcs.STACS.2023.46,
  author =	{Majewski, Konrad and Pilipczuk, Micha{\l} and Soko{\l}owski, Marek},
  title =	{{Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{46:1--46:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17698},
  URN =		{urn:nbn:de:0030-drops-176981},
  doi =		{10.4230/LIPIcs.STACS.2023.46},
  annote =	{Keywords: feedback vertex set, CMSO₂ formula, data structure, dynamic graphs, fixed-parameter tractability}
}

Keywords: feedback vertex set, CMSO₂ formula, data structure, dynamic graphs, fixed-parameter tractability
Collection: 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)
Issue Date: 2023
Date of publication: 03.03.2023


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