Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fΓΌr Informatik GmbH scholarly article en Bandyapadhyay, Sayan; Lochet, William; Lokshtanov, Daniel; Saurabh, Saket; Xue, Jie https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs

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Abstract

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set π’Ÿ of n unit disks inducing a unit-disk graph G_π’Ÿ and a number p ∈ [n], one can partition π’Ÿ into p subsets π’Ÿβ‚,… ,π’Ÿ_p such that for every i ∈ [p] and every π’Ÿ' βŠ† π’Ÿ_i, the graph obtained from G_π’Ÿ by contracting all edges between the vertices in π’Ÿ_i $1π’Ÿ' admits a tree decomposition in which each bag consists of O(p+|π’Ÿ'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22].
By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} β‹… n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} β‹… n^{O(1)} time assuming the ETH.

BibTeX - Entry

@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2022.11,
  author =	{Bandyapadhyay, Sayan and Lochet, William and Lokshtanov, Daniel and Saurabh, Saket and Xue, Jie},
  title =	{{True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16019},
  URN =		{urn:nbn:de:0030-drops-160190},
  doi =		{10.4230/LIPIcs.SoCG.2022.11},
  annote =	{Keywords: unit-disk graphs, tree decomposition, contraction decomposition, bipartization}
}

Keywords: unit-disk graphs, tree decomposition, contraction decomposition, bipartization
Seminar: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue date: 2022
Date of publication: 01.06.2022


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