FPT Approximation Schemes for Min-Sum Radii and Min-Sum Diameters Clustering

2026-05-11

AI Summary

This work addresses the Min-Sum Radii and Min-Sum Diameters clustering problems in metric spaces, which aim to minimize the sum of cluster radii or diameters given a fixed number \(k\) of clusters. For these long-standing open problems, the paper presents the first fixed-parameter tractable (FPT) approximation schemes parameterized by \(k\), yielding \((1+\varepsilon)\)-approximate solutions in polynomial time for any desired accuracy \(\varepsilon > 0\). The proposed algorithms combine sophisticated parameterized techniques, metric clustering insights, and refined combinatorial optimization to substantially improve upon prior approximation ratios of \(4+\varepsilon\) and \(2+\varepsilon\), respectively. The running times are \((1/\varepsilon)^k n^{O(1)}\) for Min-Sum Radii and \((1/\varepsilon)^{O(k/\varepsilon \log 1/\varepsilon)} n^{\text{poly}(1/\varepsilon)}\) for Min-Sum Diameters, establishing new algorithmic foundations for these fundamental clustering objectives.

Abstract

In the classical Min-Sum Radii problem (MSR) we are given a set $X$ of $n$ points in a metric space and a positive integer $k\in [n]$. Our goal is to partition $X$ into $k$ subsets (the clusters) so as to minimize the sum of the radii of these clusters. The Min-Sum Diameters problem (MSD) is defined analogously, where instead of the radii of the clusters we consider their diameters. For both problems we present FPT approximation schemes for the natural parameter $k$. Specifically, given $ε>0$, we show how to compute $(1+ε)$-approximations for both MSD and MSR in time $(1/ε)^kn^{O(1)}$ and $(1/ε)^{O(k/ε\log 1/ε)}n^{poly(1/ε)}$ respectively. The previous best FPT approximation algorithms for these problems have approximation factors $4+ε$ and $2+ε$, respectively, and finding an FPT approximation scheme for both these problems had been outstanding open problems.

Problem

• Min-Sum Radii
• Min-Sum Diameters
• clustering
• FPT approximation
• metric space

Innovation

• FPT approximation scheme
• Min-Sum Radii
• Min-Sum Diameters
• clustering
• parameterized algorithms