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：带容量约束的聚类是多年来备受关注的基础问题。本文首次给出了在一般度量空间中，将点聚类为$k$个簇以最小化簇半径总和问题（满足非均匀硬容量约束）的FPT常数因子近似算法。具体而言，我们提出了一种$(15+\epsilon)$-近似算法，运行时间为$2^{0(k^2\log k)}\cdot n^3$。当容量为均匀时，我们获得以下改进的近似界：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与Varadarajan[ESA 2020]的FPT 28-近似结果；在欧氏空间中，运行时间为$2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$的$(2+\epsilon)$-近似算法与运行时间为$2^{O(kd\log ((k/\epsilon)))}n^{3}$的$(1+\epsilon)$-近似算法；以及在允许容量违反$(1+\epsilon)$-因子的条件下，欧氏空间中运行时间为$2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$的$(1+\epsilon)$-近似算法。同时，我们通过证明在禁止任何容量违反的情况下，不存在运行时间为$f(k)\cdot n^{O(1)}$的$(1+\epsilon)$-近似算法，对上述结果进行了补充。