The $k$-center problem is a fundamental clustering objective that has been extensively studied in approximation algorithms. Recent work has sought to incorporate modern constraints such as fairness and robustness, motivated by biased and noisy data. In this paper, we study fair $k$-center with outliers, where centers must respect group-based representation constraints while up to $z$ points may be discarded. While a bi-criteria FPT approximation was previously known, no true approximation algorithm was available for this problem. We present the first deterministic $3$-approximation algorithm running in fixed-parameter tractable time parameterized by $k$. Our approach departs from projection-based methods and instead directly constructs a fair solution using a novel iterative ball-finding framework, based on a structural trichotomy that enables fixed-parameter approximation for the problem. We further extend our algorithm to fair $k$-supplier with outliers and to the more general fair-range setting with both lower and upper bounds. Finally, we show that improving the approximation factor below $3$ is $\mathrm{W[2]}$-hard, establishing the optimality of our results.

翻译：$k$-中心问题是近似算法领域中广泛研究的基础聚类目标。近期研究致力于结合公平性与鲁棒性等现代约束，以应对数据偏差与噪声问题。本文研究含离群点的公平$k$-中心问题，其中中心点需满足基于群体的代表性约束，同时允许丢弃最多$z$个数据点。尽管此前已知该问题存在双准则FPT近似算法，但尚未出现真正的近似算法。我们提出了首个确定性$3$-近似算法，其运行时间以$k$为参数的固定参数可处理时间。我们的方法摒弃了基于投影的技术，转而通过新颖的迭代球搜索框架直接构建公平解，该框架基于能够实现固定参数近似的结构三分法定理。我们进一步将算法扩展至含离群点的公平$k$-供应商问题，以及具有上下界约束的更一般公平范围设定。最后，我们证明将近似因子改进至$3$以下是$\mathrm{W[2]}$-困难的，从而确立了所提结果的最优性。