Hierarchical Clusterings of Unweighted Graphs

TL;DR

This paper investigates the complexity of hierarchical clustering in unweighted graphs, proving NP-completeness generally, but identifying specific graph classes like co-bipartite and cycle on 6 vertices as min-well-behaved, enabling efficient clustering.

Contribution

Introduces the normalization procedure for improving clusterings and characterizes min-well-behaved graphs, including co-bipartite and certain cycles, for hierarchical clustering.

Findings

Hierarchical clustering problem is NP-complete in general.

Normalization procedure can iteratively improve clusterings.

Co-bipartite graphs and cycle on 6 vertices are min-well-behaved.

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.