We consider the capacitated clustering problem in general metric spaces where the goal is to identify $k$ clusters and minimize the sum of the radii of the clusters (we call this the Capacitated-$k$-sumRadii problem). We are interested in fixed-parameter tractable (FPT) approximation algorithms where the running time is of the form $f(k) \cdot \text{poly}(n)$, where $f(k)$ can be an exponential function of $k$ and $n$ is the number of points in the input. In the uniform capacity case, Bandyapadhyay et al. recently gave a $4$-approximation algorithm for this problem. Our first result improves this to an FPT $3$-approximation and extends to a constant factor approximation for any $L_p$ norm of the cluster radii. In the general capacities version, Bandyapadhyay et al. gave an FPT $15$-approximation algorithm. We extend their framework to give an FPT $(4 + \sqrt{13})$-approximation algorithm for this problem. Our framework relies on a novel idea of identifying approximations to optimal clusters by carefully pruning points from an initial candidate set of points. This is in contrast to prior results that rely on guessing suitable points and building balls of appropriate radii around them. On the hardness front, we show that assuming the Exponential Time Hypothesis, there is a constant $c > 1$ such that any $c$-approximation algorithm for the non-uniform capacity version of this problem requires running time $2^{\Omega \left(\frac{k}{polylog(k)} \right)}$.

翻译：我们考虑一般度量空间中的容量约束聚类问题，其目标是在识别出$k$个簇的同时最小化各簇半径之和（称该问题为容量约束$k$最小半径和问题）。我们关注固定参数可解（FPT）近似算法，其运行时间形如$f(k) \cdot \text{poly}(n)$，其中$f(k)$可为$k$的指数函数，$n$为输入数据点的数量。在均匀容量情形下，Bandyapadhyay等人近期给出了该问题的$4$-近似算法。我们的第一个结果将其改进为FPT $3$-近似算法，并推广至任意$L_p$范数下簇半径的常数因子近似。在一般容量版本中，Bandyapadhyay等人提出了FPT $15$-近似算法。我们扩展其框架，给出了该问题的FPT $(4 + \sqrt{13})$-近似算法。该框架的核心创新在于：通过从初始候选点集中精心修剪数据点来识别最优簇的近似解——这与先前依赖猜测合适点并在其周围构建适当半径球体的方法形成鲜明对比。在难度下界方面，我们证明：假设指数时间假说成立，则存在常数$c > 1$使得非均匀容量版本该问题的任何$c$-近似算法都需要$2^{\Omega \left(\frac{k}{polylog(k)} \right)}$的运行时间。