License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.IPEC.2022.3
URN: urn:nbn:de:0030-drops-173597
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Bakkane, Brage I. K. ; Jaffke, Lars

On the Hardness of Generalized Domination Problems Parameterized by Mim-Width

LIPIcs-IPEC-2022-3.pdf (0.9 MB)


For nonempty σ, ρ ⊆ ℕ, a vertex set S in a graph G is a (σ, ρ)-dominating set if for all v ∈ S, |N(v) ∩ S| ∈ σ, and for all v ∈ V(G) ⧵ S, |N(v) ∩ S| ∈ ρ. The Min/Max (σ,ρ)-Dominating Set problems ask, given a graph G and an integer k, whether G contains a (σ, ρ)-dominating set of size at most k and at least k, respectively. This framework captures many well-studied graph problems related to independence and domination. Bui-Xuan, Telle, and Vatshelle [TCS 2013] showed that for finite or co-finite σ and ρ, the Min/Max (σ,ρ)-Dominating Set problems are solvable in XP time parameterized by the mim-width of a given branch decomposition of the input graph. In this work we consider the parameterized complexity of these problems and obtain the following: For minimization problems, we complete several scattered W[1]-hardness results in the literature to a full dichotomoy into polynomial-time solvable and W[1]-hard cases, and for maximization problems we obtain the same result under the additional restriction that σ and ρ are finite sets. All W[1]-hard cases hold assuming that a linear branch decomposition of bounded mim-width is given, and with the solution size being an additional part of the parameter. Furthermore, for all W[1]-hard cases we also rule out f(w)n^o(w/log w)-time algorithms assuming the Exponential Time Hypothesis, where f is any computable function, n is the number of vertices and w the mim-width of the given linear branch decomposition of the input graph.

BibTeX - Entry

  author =	{Bakkane, Brage I. K. and Jaffke, Lars},
  title =	{{On the Hardness of Generalized Domination Problems Parameterized by Mim-Width}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-173597},
  doi =		{10.4230/LIPIcs.IPEC.2022.3},
  annote =	{Keywords: generalized domination, linear mim-width, W\lbrack1\rbrack-hardness, Exponential Time Hypothesis}

Keywords: generalized domination, linear mim-width, W[1]-hardness, Exponential Time Hypothesis
Collection: 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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