License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.47
URN: urn:nbn:de:0030-drops-127139
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12713/
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Høgemo, Svein ; Paul, Christophe ; Telle, Jan Arne

Hierarchical Clusterings of Unweighted Graphs

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Abstract

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph G and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs H having the property that for any k the graph H^{(k)} being the join of k copies of H has an optimal hierarchical clustering that splits each copy of H in the same optimal way. To optimally cluster such a graph H^{(k)} we thus only need to optimally cluster the smaller graph H. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

BibTeX - Entry

@InProceedings{hgemo_et_al:LIPIcs:2020:12713,
  author =	{Svein H{\o}gemo and Christophe Paul and Jan Arne Telle},
  title =	{{Hierarchical Clusterings of Unweighted Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{47:1--47:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12713},
  URN =		{urn:nbn:de:0030-drops-127139},
  doi =		{10.4230/LIPIcs.MFCS.2020.47},
  annote =	{Keywords: Hierarchical Clustering}
}

Keywords: Hierarchical Clustering
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020

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