Fair Clustering with Minimum Representation Constraints

TL;DR

This paper introduces a novel approach to fair clustering that ensures minimum group representation in multiple clusters, using a mixed-integer optimization formulation and heuristic algorithms to handle NP-hardness, demonstrated on benchmark datasets.

Contribution

It formulates fair clustering with minimum representation constraints as a mixed-integer nonlinear problem and proposes MiniReL, an alternating minimization algorithm with heuristics for practical large-scale application.

Findings

Fair clustering achieved without increased cost

Heuristic strategies enable scalability to large datasets

Numerical results validate effectiveness on benchmarks

Abstract

Clustering is a well-studied unsupervised learning task that aims to partition data points into a number of clusters. In many applications, these clusters correspond to real-world constructs (e.g., electoral districts, playlists, TV channels), where a group (e.g., social or demographic) benefits only if it reaches a minimum level of representation in the cluster (e.g., 50% to elect their preferred candidate). In this paper, we study the k-means and k-medians clustering problems under the additional fairness constraint that each group must attain a minimum level of representation in at least a specified number of clusters. We formulate this problem as a mixed-integer (nonlinear) optimization problem and propose an alternating minimization algorithm, called MiniReL, to solve it. Although incorporating fairness constraints results in an NP-hard assignment problem within the MiniReL algorithm, we present several heuristic strategies that make the approach practical even for large datasets. Numerical results demonstrate that our method yields fair clusters without increasing clustering cost across standard benchmark datasets.