Diversity maximization in doubling metrics

TL;DR

This paper develops polynomial-time approximation schemes for diversity maximization problems in doubling metric spaces, addressing open questions and extending results to distance powers, with a proof of NP-hardness for a specific case.

Contribution

The paper introduces the first PTAS for remote-clique, remote-star, and remote-bipartition in doubling metrics, and extends analysis to distance powers, also proving NP-hardness for squared distances.

Findings

PTAS for three diversity measures in doubling metrics

Extension to distance powers q ≥ 1

NP-hardness proof for remote-clique with squared distances

Abstract

Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space $(X,d)$ and a parameter $k \in \mathbb{N}$, find a subset of $k$ elements of $X$ that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition. Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open. Our results also hold in the setting where the distances are raised to a fixed power $q\geq 1$, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the distances. Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.