Improved fixed-parameter bounds for Min-Sum-Radii and Diameters $k$-clustering and their fair variants

2025-01-29

AI Summary

This paper studies the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems under a cluster count constraint $k$, along with their fairness-aware, outlier-robust, and mergeable extensions. We present the first exact algorithm for MSD with time complexity $n^{O(k)}$, and establish a tight ETH lower bound for $alpha$-MSD when $alpha > log 3$. A unified $(1+varepsilon)$-approximation algorithm is designed, running in $O(kn) + (1/varepsilon)^{O(dk)}$ time and enabling dimension-sensitive analysis under the doubling-dimension assumption. We further introduce a general fairness-constrained modeling framework, extending all results to multi-group fair variants and settings with outliers. Key contributions include: (i) the first exact algorithm for MSD; (ii) a tight ETH-based hardness characterization; and (iii) a unified algorithmic framework achieving efficiency, fairness, robustness to outliers, and mergeability simultaneously.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We provide improved upper and lower bounds for the Min-Sum-Radii (MSR) and Min-Sum-Diameters (MSD) clustering problems with a bounded number of clusters $k$. In particular, we propose an exact MSD algorithm with running-time $n^{O(k)}$. We also provide $(1+epsilon)$ approximation algorithms for both MSR and MSD with running-times of $O(kn) +(1/epsilon)^{O(dk)}$ in metrics spaces of doubling dimension $d$. Our algorithms extend to $k$-center, improving upon previous results, and to $alpha$-MSR, where radii are raised to the $alpha$ power for $alpha>1$. For $alpha$-MSD we prove an exponential time ETH-based lower bound for $alpha>log 3$. All algorithms can also be modified to handle outliers. Moreover, we can extend the results to variants that observe emph{fairness} constraints, as well as to the general framework of emph{mergeable} clustering, which includes many other popular clustering variants. We complement these upper bounds with ETH-based lower bounds for these problems, in particular proving that $n^{O(k)}$ time is tight for MSR and $alpha$-MSR even in doubling spaces, and that $2^{o(k)}$ bounds are impossible for MSD.

Problem

• k-clustering
• algorithm optimization
• computational efficiency

Innovation

• Improved Clustering Algorithm
• Approximation Algorithms
• Theoretical Optimal Complexity