In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to the balls in $\mathcal{B}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(\log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(\log |P|)$ even with $\beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.

中文翻译：

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公制电容覆盖问题的改进边界

在度量电容覆盖 (MCC) 问题中，给定度量空间 $P$ 中的一组球 $\mathcal{B}$，具有度量 $d$ 和容量参数 $U$，目标是找到最小尺寸子集 $\mathcal{B}'\subseteq \mathcal{B}$ 以及将 $P$ 中的点分配给 $\mathcal{B}'$ 中的球，使得每个点都分配给包含它的球每个球最多分配 $U$ 点。MCC 使用贪心算法实现了 $O(\log |P|)$ 近似。另一方面，即使球的 $\beta < 3$ 因子展开，也很难在 $o(\log |P|)$ 的因子内近似。Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] 表明，对于球的因子展开为 6.47 美元的问题，可以得到一个 0(1) 美元的近似值。他们的工作留下的一个悬而未决的问题是缩小下限 $3$ 和上限 $6.47$ 之间的差距。在当前的工作中，我们表明可以通过球的仅 $4.24$ 因子展开来获得 $O(1)$-近似值。我们还展示了类似的 5 美元上限，用于更通用的 MCC 版本，此前已知的最佳上限为 9 美元。