We present a unified framework that yields EPASes for constrained $(k,z)$-clustering in metric spaces of bounded (algorithmic) scatter dimension, a notion introduced by Abbasi et al. (FOCS 2023). They showed that several well known metric families, including continuous Euclidean spaces, bounded doubling spaces, planar metrics, and bounded treewidth metrics, have bounded scatter dimension. Subsequently, Bourneuf and Pilipczuk (SODA 2025) proved that this also holds for metrics induced by graphs from any fixed proper minor closed class. Our result, in particular, addresses a major open question of Abbasi et al., whose approach to $k$-clustering in such metrics was inherently limited to \emph{Voronoi-based} objectives, where each point is connected only to its nearest chosen center. As a consequence, we obtain EPASes for several constrained clustering problems, including capacitated and matroid $(k,z)$-clustering, fault tolerant and fair $(k,z)$-clustering, as well as for metrics of bounded highway dimension. In particular, our results on capacitated and fair $k$-Median and $k$-Means provide the first EPASes for these problems across broad families of structured metrics. Previously, such results were known only in continuous Euclidean spaces, due to the works of Cohen-Addad and Li (ICALP 2019) and Bandyapadhyay, Fomin, and Simonov (ICALP 2021; JCSS 2024), respectively. Along the way, we also obtain faster EPASes for uncapacitated $k$-Median and $k$-Means, improving upon the running time of the algorithm by Abbasi et al. (FOCS 2023).

翻译：我们提出了一个统一框架，该框架可为具有有界（算法）散射维度的度量空间中的约束$(k,z)$-聚类问题生成高效参数化近似方案（EPAS），其中散射维度的概念由 Abbasi 等人（FOCS 2023）引入。他们证明了包括连续欧几里得空间、有界倍维空间、平面度量以及有界树宽度量在内的多个著名度量族都具有有界散射维度。随后，Bourneuf 和 Pilipczuk（SODA 2025）证明了这一性质同样适用于由任何固定真闭子图类诱导的图度量。我们的结果特别解决了 Abbasi 等人提出的一个主要开放性问题，他们针对此类度量中$k$-聚类的方法本质上局限于\emph{基于Voronoi}的目标函数，即每个点仅连接到其最近的选择中心。因此，我们为多个约束聚类问题获得了EPAS，包括容量约束和拟阵$(k,z)$-聚类、容错和公平$(k,z)$-聚类，以及有界公路维度的度量。特别地，我们在容量约束和公平$k$-中位数与$k$-均值问题上的结果，为这些广泛的结构化度量族提供了首个EPAS。此前，由于 Cohen-Addad 和 Li（ICALP 2019）以及 Bandyapadhyay、Fomin 和 Simonov（ICALP 2021；JCSS 2024）的工作，此类结果仅在连续欧几里得空间中已知。在此过程中，我们还为非容量约束的$k$-中位数和$k$-均值问题获得了更快的EPAS，改进了 Abbasi 等人（FOCS 2023）算法的运行时间。