Coreset Strikes Back: Improved Parameterized Approximation Schemes for (Constrained) k-Median/Means

2025-04-09

AI Summary

This paper investigates the existence of an efficient parameterized approximation scheme (EPAS) for the capacitated and fair $k$-Median/Means clustering problems in metric spaces with bounded algorithmic scatter dimension. Addressing the limitation of prior approaches—which only handle unconstrained nearest-center assignments—we propose the first unified EPAS framework supporting both capacity and fairness constraints. Our core technique constructs a coreset of size $(k log n / varepsilon)^{O(1)}$, leveraging algorithmic scatter dimension theory and constrained optimization modeling. Our contributions include: (i) the first EPAS for capacitated and fair $k$-Median/Means in broad classes of metric spaces—including fixed minor-free graph families, Euclidean spaces, planar graphs, and low-treewidth graphs; and (ii) significantly improved running time for the unconstrained case, outperforming the state-of-the-art result from FOCS’23.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Algorithmic scatter dimension is a notion of metric spaces introduced recently by Abbasi et al. (FOCS 2023), which unifies many well-known metric spaces, including continuous Euclidean space, bounded doubling space, planar and bounded treewidth metrics. Recently, Bourneuf and Pilipczuk (SODA 2025) showed that metrics induced by graphs from any fixed proper minor closed graph class have bounded scatter dimension. Abbasi et al. presented a unified approach to obtain EPASes (i.e., $(1+epsilon)$-approximations running in time FPT in $k$ and $epsilon$) for $k$-Clustering in metrics of bounded scatter dimension. However, a seemingly inherent limitation of their approach was that it could only handle clustering objectives where each point was assigned to the closest chosen center. They explicitly asked, if there exist EPASes for constrained $k$-Clustering in metrics of bounded scatter dimension. We present a unified framework which yields EPASes capacitated and fair $k$-Median/Means in metrics of bounded algorithmic scatter dimension. Our framework exploits coresets for such constrained clustering problems in a novel manner, and notably requires only coresets of size $(klog n/epsilon)^{O(1)}$, which are usually constuctible even in general metrics. Note that due to existing lower bounds it is impossible to obtain such an EPAS for Capacitated $k$-Center, thus essentially answering the complete spectrum of the question. Our results on capacitated and fair $k$-Median/Means provide the first EPASes for these problems in broad families of metric spaces. Earlier such results were only known in continuous Euclidean spaces due to Cohen-Addad&Li, (ICALP 2019), and Bandyapadhyay, Fomin&Simonov, (ICALP 2021; JCSS 2024), respectively. Along the way, we obtain faster EPASes for uncapacitated $k$-Median/Means, improving upon the running time of the algorithm by Abbasi et al.

Problem

• Develop EPAS for constrained k-Clustering in bounded scatter dimension metrics
• Unify framework for capacitated and fair k-Median/Means problems
• Improve EPAS speed for uncapacitated k-Median/Means clustering

Innovation

• Unified framework for constrained k-Clustering
• Novel coreset exploitation in bounded scatter metrics
• Improved EPASes for capacitated and fair clustering