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.MFCS.2023.61
URN: urn:nbn:de:0030-drops-185958
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18595/
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Lafond, Manuel ; Luo, Weidong

Parameterized Complexity of Domination Problems Using Restricted Modular Partitions

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LIPIcs-MFCS-2023-61.pdf (0.8 MB)


Abstract

For a graph class 𝒢, we define the 𝒢-modular cardinality of a graph G as the minimum size of a vertex partition of G into modules that each induces a graph in 𝒢. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if 𝒢 has bounded modular-width, the W[1]-hardness of a problem in 𝒢-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic (α, β)-domination problem, which is a generalization of known domination problems such as Bounded Degree Deletion, k-Domination, and α-Domination. We show that for graph classes 𝒢 that require arbitrarily large solution tables, these problems are W[1]-hard in the 𝒢-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and cluster-modular cardinality.

BibTeX - Entry

@InProceedings{lafond_et_al:LIPIcs.MFCS.2023.61,
  author =	{Lafond, Manuel and Luo, Weidong},
  title =	{{Parameterized Complexity of Domination Problems Using Restricted Modular Partitions}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{61:1--61:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18595},
  URN =		{urn:nbn:de:0030-drops-185958},
  doi =		{10.4230/LIPIcs.MFCS.2023.61},
  annote =	{Keywords: modular-width, parameterized algorithms, W-hardness, 𝒢-modular cardinality}
}

Keywords: modular-width, parameterized algorithms, W-hardness, 𝒢-modular cardinality
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023


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