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Hierarchical Clusterings of Unweighted Graphs

Svein Hogemo 1 Christophe Paul 2 Jan Arne Telle 1 
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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.
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Submitted on : Friday, November 27, 2020 - 10:59:46 AM
Last modification on : Friday, August 5, 2022 - 3:02:53 PM
Long-term archiving on: : Sunday, February 28, 2021 - 6:59:14 PM


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Svein Hogemo, Christophe Paul, Jan Arne Telle. Hierarchical Clusterings of Unweighted Graphs. MFCS 2020 - 45th International Symposium on Mathematical Foundations of Computer Science, Aug 2020, Prague, Czech Republic. pp.47:1-47:13, ⟨10.4230/LIPIcs.MFCS.2020.47⟩. ⟨lirmm-03027532⟩



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