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.SWAT.2022.29
URN: urn:nbn:de:0030-drops-161891
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16189/
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Kunz, Pascal ; Fluschnik, Till ; Niedermeier, Rolf ; Renken, Malte

Most Classic Problems Remain NP-Hard on Relative Neighborhood Graphs and Their Relatives

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Abstract

Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the following classes of proximity graphs: relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the aforementioned problems remain NP-hard on these graphs, except for 3-Colorability and Hamiltonian Cycle on relatively closest graphs, where the former is trivial and the latter is left open. Moreover, for every NP-hard case we additionally show that no 2^{o(n^{1/4})}-time algorithm exists unless the Exponential-Time Hypothesis (ETH) fails, where n denotes the number of vertices.

BibTeX - Entry

@InProceedings{kunz_et_al:LIPIcs.SWAT.2022.29,
  author =	{Kunz, Pascal and Fluschnik, Till and Niedermeier, Rolf and Renken, Malte},
  title =	{{Most Classic Problems Remain NP-Hard on Relative Neighborhood Graphs and Their Relatives}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{29:1--29:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16189},
  URN =		{urn:nbn:de:0030-drops-161891},
  doi =		{10.4230/LIPIcs.SWAT.2022.29},
  annote =	{Keywords: Proximity Graphs, Relatively Closest Graphs, Gabriel Graphs, 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, Independent Set, Exponential-Time Hypothesis}
}

Keywords: Proximity Graphs, Relatively Closest Graphs, Gabriel Graphs, 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, Independent Set, Exponential-Time Hypothesis
Collection: 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)
Issue Date: 2022
Date of publication: 22.06.2022

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