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

TL;DR

This paper develops fixed-parameter tractable approximation schemes for Min-Sum Radii and Min-Sum Diameters clustering problems, achieving near-optimal solutions efficiently for small parameter values.

Contribution

It introduces the first FPT approximation schemes for both problems, improving upon previous approximation algorithms and resolving open problems.

Findings

Achieves (1+ε)-approximations in FPT time for both problems.

Improves previous approximation factors from 4+ε and 2+ε to near-optimal schemes.

Provides algorithms with running times exponential in k but polynomial in input size.

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 $\epsilon>0$, we show how to compute $(1+\epsilon)$-approximations for both MSD and MSR in time $(1/\epsilon)^kn^{O(1)}$ and $(1/\epsilon)^{O(k/\epsilon \log 1/\epsilon)}n^{poly(1/\epsilon)}$ respectively. The previous best FPT approximation algorithms for these problems have approximation factors $4+\epsilon$ and $2+\epsilon$, respectively, and finding an FPT approximation scheme for both these problems had been outstanding open problems.