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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.42
URN: urn:nbn:de:0030-drops-144828
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14482/
Duarte, Gabriel L. ;
de Oliveira Oliveira, Mateus ;
Souza, Uéverton S.
Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances
Abstract
Clique-width and treewidth are two of the most important and useful graph parameters, and several problems can be solved efficiently when restricted to graphs of bounded clique-width or treewidth. Bounded treewidth implies bounded clique-width, but not vice versa. Problems like Longest Cycle, Longest Path, MaxCut, Edge Dominating Set, and Graph Coloring are fixed-parameter tractable when parameterized by the treewidth, but they cannot be solved in FPT time when parameterized by the clique-width unless FPT = W[1], as shown by Fomin, Golovach, Lokshtanov, and Saurabh [SIAM J. Comput. 2010, SIAM J. Comput. 2014]. For a given problem that is fixed-parameter tractable when parameterized by treewidth, but intractable when parameterized by clique-width, there may exist infinite families of instances of bounded clique-width and unbounded treewidth where the problem can be solved efficiently. In this work, we initiate a systematic study of the parameters co-treewidth (the treewidth of the complement of the input graph) and co-degeneracy (the degeneracy of the complement of the input graph). We show that Longest Cycle, Longest Path, and Edge Dominating Set are FPT when parameterized by co-degeneracy. On the other hand, Graph Coloring is para-NP-complete when parameterized by co-degeneracy but FPT when parameterized by the co-treewidth. Concerning MaxCut, we give an FPT algorithm parameterized by co-treewidth, while we leave open the complexity of the problem parameterized by co-degeneracy. Additionally, we show that Precoloring Extension is fixed-parameter tractable when parameterized by co-treewidth, while this problem is known to be W[1]-hard when parameterized by treewidth. These results give evidence that co-treewidth is a useful width parameter for handling dense instances of problems for which an FPT algorithm for clique-width is unlikely to exist. Finally, we develop an algorithmic framework for co-degeneracy based on the notion of Bondy-Chvátal closure.
BibTeX - Entry
@InProceedings{duarte_et_al:LIPIcs.MFCS.2021.42,
author = {Duarte, Gabriel L. and de Oliveira Oliveira, Mateus and Souza, U\'{e}verton S.},
title = {{Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances}},
booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
pages = {42:1--42:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-201-3},
ISSN = {1868-8969},
year = {2021},
volume = {202},
editor = {Bonchi, Filippo and Puglisi, Simon J.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14482},
URN = {urn:nbn:de:0030-drops-144828},
doi = {10.4230/LIPIcs.MFCS.2021.42},
annote = {Keywords: FPT, treewidth, degeneracy, complement graph, Bondy-Chv\'{a}tal closure}
}
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Keywords: |
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FPT, treewidth, degeneracy, complement graph, Bondy-Chvátal closure |
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Collection: |
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46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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18.08.2021 |
