FPT Approximations for Fair Sum of Radii with Outliers and General Norm Objectives

Abstract

The sum of radii problem is a classical clustering problem in which, given a set $X$ of points and an integer $k$, the goal is to place $k$ balls that cover $X$ while minimizing the sum of their radii. Recent work has focused on incorporating modern constraints such as fairness and robustness, motivated by biased and noisy data. We study the fair sum of radii with outliers problem, where the chosen centers must satisfy group-based representation constraints while allowing up to $z$ points to be excluded. We present a $(3+\epsilon)$-approximation algorithm that runs in fixed-parameter tractable time parameterized by $k$. Our framework extends to the more general setting where the objective is a monotone symmetric norm of the radii, achieving a $(3+\epsilon)$-approximation for any fixed norm; this guarantee is tight under Gap-ETH. Moreover, the algorithm is oblivious to the choice of norm: it outputs a small list of candidate solutions such that, for every monotone symmetric norm $f$, the list contains a $(3+\epsilon)$-approximate solution under $f$. Our approach is based on a novel iterative ball-finding framework that uncovers a structural trichotomy in the optimal clustering, enabling us to directly construct fair solutions while handling outliers. Finally, we extend our techniques to the more general fair-range setting, where each group is subject to both lower and upper bounds.