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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.IPEC.2022.3
URN: urn:nbn:de:0030-drops-173597
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17359/
Bakkane, Brage I. K. ;
Jaffke, Lars
On the Hardness of Generalized Domination Problems Parameterized by Mim-Width
Abstract
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
@InProceedings{bakkane_et_al:LIPIcs.IPEC.2022.3,
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 = {https://drops.dagstuhl.de/opus/volltexte/2022/17359},
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}
}
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Keywords: |
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generalized domination, linear mim-width, W[1]-hardness, Exponential Time Hypothesis |
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Collection: |
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17th International Symposium on Parameterized and Exact Computation (IPEC 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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14.12.2022 |
