Hierarchical Clusterings of Unweighted Graphs

Authors Svein Høgemo, Christophe Paul, Jan Arne Telle

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Author Details

Svein Høgemo
  • Department of Informatics, University of Bergen, Norway
Christophe Paul
  • LIRMM, University of Montpellier, CNRS, France
Jan Arne Telle
  • Department of Informatics, University of Bergen, Norway

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Svein Høgemo, Christophe Paul, and Jan Arne Telle. Hierarchical Clusterings of Unweighted Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Hierarchical Clustering


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