FPT approximations for Capacitated Sum of Radii and Diameters

2024-09-08

arXiv.org

AI Summary

This paper studies the capacitated $k$-clustering problem: given $n$ points and a capacity $U_p$ for each potential center $p$, partition the points into $k$ clusters such that each cluster’s size does not exceed its center’s capacity, and the sum of cluster radii (or diameters) is minimized. We first prove that the problem is APX-hard in general metric spaces and, under the gap-ETH assumption, admits no fixed-parameter tractable approximation scheme (FPT-AS). We then present the first FPT algorithm with approximation ratio $approx 5.83$, improving upon the previous best ratio of $approx 7.61$. Our framework is further extended to handle monotone symmetric norm objectives and the uniform-capacity variant. Technically, we integrate parameterized algorithm design, geometric clustering analysis, and tight approximation-ratio analysis, yielding a unified approach with strong theoretical guarantees and broad applicability.

Technology Category

• Machine Learning: Clustering
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The Capacitated Sum of Radii problem involves partitioning a set of points $P$, where each point $pin P$ has capacity $U_p$, into $k$ clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point $p$ is at most $U_p$. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a $approx5.83$-approximation algorithm in FPT time (improving a previous $approx7.61$ approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.

Problem

• Capacitated Sum of Radii APX-hard with no FPT-AS
• Develop FPT approximation algorithm for cluster radii minimization
• Improve approximations for uniform capacities and diameters problems

Innovation

• FPT approximation for capacitated sum radii
• Improved 5.83 approximation algorithm
• Handles general monotone symmetric norms