The Price of Hierarchical Clustering

TL;DR

This paper investigates the limitations of hierarchical clustering by quantifying its approximation quality for specific cost functions, establishing bounds on how close hierarchical solutions can get to optimal clusterings.

Contribution

It introduces the concept of the price of hierarchy and provides improved bounds for the approximation factors in hierarchical clustering for $k$-center and $k$-diameter problems.

Findings

Upper bound on the price of hierarchy for $k$-diameter is approximately 5.83.

Exact price of hierarchy for $k$-center is proven to be 4.

Exact price of hierarchy for $k$-diameter is proven to be approximately 5.83.

Abstract

Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of $\alpha$ if the costs of each $k$-clustering in the hierarchy are at most $\alpha$ times the costs of an optimal $k$-clustering. We study as cost functions the maximum (discrete) radius of any cluster ($k$-center problem) and the maximum diameter of any cluster ($k$-diameter problem). In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of $1$ cannot be achieved. We call the smallest approximation factor that can be achieved for any instance the price of hierarchy. For the $k$-diameter problem we improve the upper bound on the price of hierarchy to $3+2\sqrt{2}\approx 5.83$. Moreover we significantly improve the lower bounds for $k$-center and $k$-diameter, proving a price of hierarchy of exactly $4$ and $3+2\sqrt{2}$, respectively.