FPT Constant Approximation Algorithms for Colorful Sum of Radii

TL;DR

This paper introduces fixed-parameter tractable algorithms that provide constant-factor approximations for the colorful sum of radii problem, a complex clustering task with outlier and color constraints, improving upon previous logarithmic approximations.

Contribution

The authors develop the first fixed-parameter tractable algorithms achieving constant-factor approximations for the colorful sum of radii problem, with improved running times and approximation guarantees.

Findings

Achieved a (2+ε)-approximation with exponential dependence on k and m.

Developed a (7+ε)-approximation algorithm with reduced dependency on m.

Provided the first constant-factor FPT algorithms for this problem.

Abstract

We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_\omega$, along with per-class outlier bounds $m_1, m_2, \dots, m_\omega$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log \omega)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$.