FPT approximations for Capacitated Sum of Radii and Diameters

TL;DR

This paper studies the Capacitated Sum of Radii problem, proving its computational hardness and providing a new faster approximation algorithm with improved approximation ratio, applicable to related problems and general norms.

Contribution

It introduces a faster FPT approximation algorithm with a better ratio and establishes hardness results, extending to general norms and related problems.

Findings

The problem is APX-hard and has no FPT-AS under gap-ETH.

An improved 5.83-approximation algorithm in FPT time.

Enhanced approximation factors for uniform capacities and related problems.

Abstract

The Capacitated Sum of Radii problem involves partitioning a set of points $P$, where each point $p\in P$ 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.