Tight FPT Approximations for Fair $k$-center with Outliers

2026-02-05

AI Summary

This work addresses the fair k-center clustering problem with outliers, where the goal is to exclude at most z outliers from noisy data while satisfying group fairness constraints and minimizing the clustering radius. The authors present the first deterministic 3-approximation algorithm that runs in fixed-parameter tractable (FPT) time. Their key contributions include introducing the first true (non-bicriteria) approximation algorithm for this setting, designing an iterative sphere-discovery framework based on a structural trisection technique that directly constructs a feasible fair solution, and proving that the 3-approximation ratio is optimal under the W[2] hardness assumption. The approach naturally extends to related problems, including fair k-supplier and fair range clustering with both lower and upper bounds.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness and ethical considerations in crowd work and in human-in-the-loop AI systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The $k$-center problem is a fundamental clustering objective that has been extensively studied in approximation algorithms. Recent work has sought to incorporate modern constraints such as fairness and robustness, motivated by biased and noisy data. In this paper, we study fair $k$-center with outliers, where centers must respect group-based representation constraints while up to $z$ points may be discarded. While a bi-criteria FPT approximation was previously known, no true approximation algorithm was available for this problem. We present the first deterministic $3$-approximation algorithm running in fixed-parameter tractable time parameterized by $k$. Our approach departs from projection-based methods and instead directly constructs a fair solution using a novel iterative ball-finding framework, based on a structural trichotomy that enables fixed-parameter approximation for the problem. We further extend our algorithm to fair $k$-supplier with outliers and to the more general fair-range setting with both lower and upper bounds. Finally, we show that improving the approximation factor below $3$ is $\mathrm{W[2]}$-hard, establishing the optimality of our results.

Problem

• fair k-center
• outliers
• approximation algorithm
• fixed-parameter tractable
• clustering

Innovation

• fair k-center
• outliers
• fixed-parameter tractable
• iterative ball-finding
• approximation algorithm