Hierarchical Clusterings of Unweighted Graphs
| Autor: | Høgemo, Svein, Paul, Christophe, Telle, Jan Arne |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 19 pages, 7 figures. Extended version of conference paper, to appear in proceedings from MFCS 2020 |
| Databáze: | arXiv |
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