In this paper, we consider the Minimum-Load $k$-Clustering/Facility Location (MLkC) problem where we are given a set $P$ of $n$ points in a metric space that we have to cluster and an integer $k$ that denotes the number of clusters. Additionally, we are given a set $F$ of cluster centers in the same metric space. The goal is to select a set $C\subseteq F$ of $k$ centers and assign each point in $P$ to a center in $C$, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it. Although clustering/facility location problems have a rich literature, the minimum-load objective is not studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [ACM Trans. Algo. 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is $O(k)$, even in the plane. In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017], which generalizes MLkC. Here the input points are colored by one of the $\ell$ colors denoting the group they belong to. MLkC is the special case with $\ell=1$. Considering this problem, we are able to obtain a $3$-approximation in $f(k,\ell)\cdot n^{O(1)}$ time. Also, our scheme leads to an improved $(1 + \epsilon)$-approximation in case of Euclidean norm, and in this case, the running time depends only polynomially on the dimension $d$. Our results imply the same approximations for MLkC with running time $f(k)\cdot n^{O(1)}$, achieving the first constant approximations for this problem in general and Euclidean metric spaces.

中文翻译：

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公平最小负载聚类的 FPT 近似

在本文中，我们考虑了最小负载 $k$-聚类/设施位置 (MLkC) 问题，在该问题中，我们在必须聚类的度量空间中给定一组 $p$，其中包含 $n$ 个点和一个整数 $k $ 表示集群的数量。此外，我们在同一度量空间中获得了一组 $F$ 的聚类中心。目标是选择一组 $k$ 中心的 $C\subseteq F$ 并将 $P$ 中的每个点分配给 $C$ 中的一个中心，从而使所有中心的最大负载最小化。这里中心的载荷是它与分配给它的点之间的距离之和。尽管集群/设施选址问题有丰富的文献，但最小负载目标并未得到实质性研究，因此 MLkC 仍然是一个知之甚少的问题。更有趣的是，即使在一些特殊情况下，这个问题也非常困难，包括 Ahmadian 等人所示的在线指标。[ACM 翻译。算法。2018]。它们还显示了平面中问题的 APX 硬度。另一方面，MLkC 最著名的近似因子是 $O(k)$，即使在平面中也是如此。在这项工作中，我们研究了受 Chierichetti 等人工作启发的 MLkC 的公平版本。[NeurIPS，2017]，概括了 MLkC。此处输入点由表示它们所属组的 $\ell$ 颜色之一着色。MLkC 是 $\ell=1$ 的特例。考虑到这个问题，我们能够在 $f(k,\ell)\cdot n^{O(1)}$ 时间内获得 $3$-近似值。此外，我们的方案在欧几里德范数的情况下导致了改进的 $(1 + \epsilon)$ 近似，在这种情况下，运行时间仅多项式取决于维度 $d$。