FPT Approximation for Fair Minimum-Load Clustering

Sayan Bandyapadhyay, F. Fomin, P. Golovach, Nidhi Purohit, Kirill Simonov
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引用次数: 1

Abstract

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.
公平最小负载聚类的FPT近似
在本文中,我们考虑最小负载$k$聚类/设施位置(MLkC)问题,其中我们在度量空间中给定$n$点的集合$P$和一个整数$k$表示聚类的数量。另外,我们在相同的度量空间中得到了一组聚类中心。目标是选择$k$中心的集合$C\subseteq F$,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。在这里,一个中心的载荷是它和指定给它的点之间的距离的总和。尽管聚类/设施选址问题有丰富的文献,但最小负荷目标并没有实质性的研究,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,包括ahmaddian等人所示的一行指标,这个问题也是出了名的困难。藻类,2018]。它们还显示了问题在平面上的apx硬度。另一方面,MLkC最著名的近似因子是0 (k),即使在平面上也是如此。在这项工作中,我们受Chierichetti等人[NeurIPS, 2017]的工作启发,研究了MLkC的公平版本,该版本概括了MLkC。在这里,输入点用表示它们所属组的$\ell$颜色之一着色。MLkC是$\ell=1$的特殊情况。考虑到这个问题,我们能够在$f(k,\ well)\cdot n^{O(1)}$时间内得到$3$-近似。此外,我们的方案在欧几里得范数的情况下导致改进的$(1 + \epsilon)$-近似,在这种情况下,运行时间仅多项式地取决于维度$d$。我们的结果意味着对于运行时间$f(k)\cdot n^{O(1)}$的MLkC具有相同的近似,在一般和欧几里德度量空间中实现了该问题的第一个常数近似。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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