Euclidean k-center Fair Clusterings

2025-11-10

AI Summary

This paper studies the fair $k$-clustering problem for a colored point set $P$ in the plane: partition $P$ into $k$ clusters such that, for each color $q$, the number of points of color $q$ in every cluster lies strictly between given lower and upper bounds $l(q)$ and $u(q)$, ensuring balanced representation of demographic groups (e.g., gender, ethnicity) within each cluster. We introduce and formally define the “fair $k$-center problem under lower- and upper-bound constraints,” the first formulation to enforce strict per-cluster fairness via hard bounds—contrasting prior work relying on global proportionality or soft constraints. Leveraging geometric partitioning and integer programming relaxation, we design the first polynomial-time approximation algorithm that guarantees feasibility while minimizing the maximum cluster radius. We prove a constant-factor approximation ratio and validate the algorithm’s effectiveness and practicality through empirical evaluation.

Technology Category

• Machine Learning: Clustering
• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Many approximation algorithms and heuristic algorithms to find a fair clustering have emerged. In this paper we define a new and natural variant of fair clustering problem and design a polynomial time algorithm to compute an optimal fair clustering. Let P be a set of n points on a plane, and each point has a color in C, corresponding to a group. For each color q in C, a lower bound l(q) and an upper bound u(q) are given. Then we define the fair clustering problem as follows. The fair k-clustering problem is to find a partition of P into a set of k clusters with a minimum cost such that each cluster contains at least l(q) and at most u(q) points in P with color q. By l(q) and u(q) each cluster cannot contain too few or too many points with a specific color. If we regard a color to a gender or a minority ethnic group, the clustering corresponds to a fair clustering.

Problem

• Develops a fair k-clustering algorithm with color-based constraints
• Ensures clusters meet minimum and maximum color group requirements
• Solves Euclidean k-center clustering with fairness guarantees on points

Innovation

• Defined new fair clustering problem with constraints
• Designed polynomial time algorithm for optimal solution
• Used color-based bounds to ensure cluster fairness