Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+\epsilon)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $\epsilon$)-approximation with running time $2^{O(k\log(k/\epsilon))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $\epsilon$)-approximation with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ and a $(1+\epsilon)$-approximation with running time $2^{O(kd\log ((k/\epsilon)))}n^{3}$ in the Euclidean space; and a (1 + $\epsilon$)-approximation in the Euclidean space with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ if we are allowed to violate the capacities by (1 + $\epsilon$)-factor. We complement this result by showing that there is no (1 + $\epsilon$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.

翻译：容量约束聚类是一个基本问题，多年来吸引了大量关注。本文首次提出在一般度量空间中将点聚类为$k$个簇、以最小化聚类半径之和（受非均匀硬容量约束）的FPT常数因子近似算法。具体而言，我们给出一个$(15+\epsilon)$-近似算法，运行时间为$2^{0(k^2\log k)}\cdot n^3$。当容量均匀时，我们得到以下改进的近似界：运行时间为$2^{O(k\log(k/\epsilon))}n^3$的$(4+\epsilon)$-近似算法，显著改进了Inamdar和Varadarajan [ESA 2020]的FPT 28-近似算法；在欧几里得空间中，运行时间为$2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$的$(2+\epsilon)$-近似算法和运行时间为$2^{O(kd\log ((k/\epsilon)))}n^{3}$的$(1+\epsilon)$-近似算法；若允许容量违反$(1+\epsilon)$-因子，则在欧几里得空间中得到运行时间为$2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$的$(1+\epsilon)$-近似算法。我们通过证明若不允许任何容量违反，则不存在运行时间为$f(k)\cdot n^{O(1)}$的$(1+\epsilon)$-近似算法来补充这一结果。