On Dasgupta's hierarchical clustering objective and its relation to other graph parameters

TL;DR

This paper explores a new hierarchical clustering objective based on total depth, introduces a related robustness measure, and studies its computational complexity and algorithms for specific graph classes.

Contribution

It proposes a novel total-depth objective for hierarchical clustering, analyzes its properties, and provides algorithms and approximations for certain graph classes.

Findings

The new parameter measures robustness against vertex removal.

Polynomial-time algorithms are provided for caterpillars and bounded leaf trees.

A 2-approximation algorithm is developed for general trees.

Abstract

The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear-time algorithms exist for trees. Motivated by a correspondence with Dasgupta's objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for minimization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and therefore intractable like the other three are known to be. We give polynomial-time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm.