# Hierarchical Clusterings of Unweighted Graphs

arXiv: https://arxiv.org/abs/2008.03061 · HTML: https://mlsanity.com/paper/2008.03061/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2020.

## Built on by (most cited first, 1 of 1)

- [Hierarchical Clustering: New Bounds and Objective](https://mlsanity.com/paper/2111.06863/) (2021)

## Abstract (arXiv)

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.