FPT Approximation for Constrained Metric k-Median/Means

Dishant Goyal, Ragesh Jaiswal, Amit Kumar
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引用次数: 6

Abstract

The Metric $k$-median problem over a metric space $(\mathcal{X}, d)$ is defined as follows: given a set $L \subseteq \mathcal{X}$ of facility locations and a set $C \subseteq \mathcal{X}$ of clients, open a set $F \subseteq L$ of $k$ facilities such that the total service cost, defined as $\Phi(F, C) \equiv \sum_{x \in C} \min_{f \in F} d(x, f)$, is minimised. The metric $k$-means problem is defined similarly using squared distances. In many applications there are additional constraints that any solution needs to satisfy. This gives rise to different constrained versions of the problem such as $r$-gather, fault-tolerant, outlier $k$-means/$k$-median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. We give FPT algorithms with constant approximation guarantee for a range of constrained $k$-median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a $(3+\varepsilon)$-approximation and $(9+\varepsilon)$-approximation for the constrained versions of the $k$-median and $k$-means problem respectively in FPT time. In many practical settings of the $k$-median/means problem, one is allowed to open a facility at any client location, i.e., $C \subseteq L$. For this special case, our algorithm gives a $(2+\varepsilon)$-approximation and $(4+\varepsilon)$-approximation for the constrained versions of $k$-median and $k$-means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm.
约束度量k-中值/均值的FPT近似
度量空间$(\mathcal{X}, d)$上的度量$k$ -中位数问题定义如下:给定一组$L \subseteq \mathcal{X}$的设施位置和一组$C \subseteq \mathcal{X}$的客户,打开一组$F \subseteq L$的$k$设施,使总服务成本(定义为$\Phi(F, C) \equiv \sum_{x \in C} \min_{f \in F} d(x, f)$)最小化。度量$k$ -均值问题类似地使用平方距离来定义。在许多应用程序中,任何解决方案都需要满足额外的约束。这就产生了问题的不同约束版本,例如$r$ -gather、容错、outlier $k$ -means/ $k$ -median问题。令人惊讶的是,对于许多这些约束问题,没有已知的常数近似算法。针对一系列受限$k$ -中值/均值问题,给出了具有常数逼近保证的FPT算法。对于某些约束问题,我们的算法是第一个常数因子近似算法,而对于其他问题,我们改进或匹配了以前工作的近似保证。我们在Ding和Xu的统一框架内工作,这使我们能够同时获得一系列约束问题的算法。特别是,我们分别在FPT时间内获得了$k$ -median和$k$ -means问题的约束版本的$(3+\varepsilon)$ -逼近和$(9+\varepsilon)$ -逼近。在$k$ -中位数/平均值问题的许多实际设置中,允许在任何客户位置(即$C \subseteq L$)开设设施。对于这种特殊情况,我们的算法分别在FPT时间内对$k$ -median和$k$ -means问题的约束版本给出$(2+\varepsilon)$ -逼近和$(4+\varepsilon)$ -逼近。由于我们的算法基于简单的采样技术,因此它也可以转换为恒定通过对数空间流算法。
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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