Fair Minimum Representation Clustering

TL;DR

This paper introduces a fair clustering method that ensures demographic groups reach minimum representation levels in clusters, addressing unfair outcomes of traditional k-means algorithms.

Contribution

It formulates a new fair k-means clustering framework using mixed-integer optimization and proposes MiniReL, a variant of Lloyd's algorithm that enforces fairness constraints.

Findings

MiniReL produces fairer clusters with minimal cost increase.

The approach is computationally feasible for large datasets.

Numerical results demonstrate improved fairness in benchmark datasets.

Abstract

Clustering is an unsupervised learning task that aims to partition data into a set of clusters. In many applications, these clusters correspond to real-world constructs (e.g. electoral districts) whose benefit can only be attained by groups when they reach a minimum level of representation (e.g. 50\% to elect their desired candidate). This paper considers the problem of performing k-means clustering while ensuring groups (e.g. demographic groups) have that minimum level of representation in a specified number of clusters. We show that the popular $k$-means algorithm, Lloyd's algorithm, can result in unfair outcomes where certain groups lack sufficient representation past the minimum threshold in a proportional number of clusters. We formulate the problem through a mixed-integer optimization framework and present a variant of Lloyd's algorithm, called MiniReL, that directly incorporates the fairness constraints. We show that incorporating the fairness criteria leads to a NP-Hard sub-problem within Lloyd's algorithm, but we provide computational approaches that make the problem tractable for even large datasets. Numerical results show that the approach is able to create fairer clusters with practically no increase in the k-means clustering cost across standard benchmark datasets.