Preserving Diversity when Partitioning: A Geometric Approach

TL;DR

This paper introduces a geometric approach to partitioning communities to preserve diversity, providing optimal solutions for two types and discussing challenges for multiple types.

Contribution

It offers a novel geometric interpretation of Simpson's diversity metric and presents an efficient algorithm for two-type community partitioning.

Findings

Optimal partitioning preserves global diversity in certain instances.

Polynomial-time algorithm for two-type community partitioning.

Discussion of open challenges for multiple types.

Abstract

Diversity plays a crucial role in multiple contexts such as team formation, representation of minority groups and generally when allocating resources fairly. Given a community composed by individuals of different types, we study the problem of partitioning this community such that the global diversity is preserved as much as possible in each subgroup. We consider the diversity metric introduced by Simpson in his influential work that, roughly speaking, corresponds to the inverse probability that two individuals are from the same type when taken uniformly at random, with replacement, from the community of interest. We provide a novel perspective by reinterpreting this quantity in geometric terms. We characterize the instances in which the optimal partition exactly preserves the global diversity in each subgroup. When this is not possible, we provide an efficient polynomial-time algorithm that outputs an optimal partition for the problem with two types. Finally, we discuss further challenges and open questions for the problem that considers more than two types.