低维子模聚类
Submodular Clustering in Low Dimensions
摘要：
我们研究了一个聚类问题，其目标是最大限度地覆盖输入点的$k$选择的中心。具体地，给定一组$n$点$P \subseteq \mathbb{R}^d$，目标是挑选最大化到点$P$的服务$ \sum_{p \in P}\mathsfφ\bigl( \mathsf{d}(p,C) \bigr) $的$k$中心$C \subseteq \mathbb{R}^d$，其中$\mathsf{d}(p,C)$是$p$到其在$C$中的最近中心的距离，并且$\mathsfφ$是非递增服务函数$\mathsfφ : \mathbb{R}^+ \to \mathbb{R}^+$。这包括放置$k$基站以使到客户端的总带宽最大化的问题-实际上，客户端离其最近的基站越近，它可以发送/接收的数据就越多，并且目标是放置$k$基站以使总带宽最大化。我们提供了一个$n^{\varepsilon^{-O(d)}}$时间算法，实现了$(1-\varepsilon)$ -近似的这个问题。值得注意的是，运行时不依赖于参数$k$，并且它适用于任意非递增服务函数$\mathsfφ : \mathbb{R}^+ \to \mathbb{R}^+$。
英文摘要：
We study a clustering problem where the goal is to maximize the coverage of the input points by $k$ chosen centers. Specifically, given a set of $n$ points $P \subseteq \mathbb{R}^d$, the goal is to pick $k$ centers $C \subseteq \mathbb{R}^d$ that maximize the service $ \sum_{p \in P}\mathsfφ\bigl( \mathsf{d}(p,C) \bigr) $ to the points $P$, where $\mathsf{d}(p,C)$ is the distance of $p$ to its nearest center in $C$, and $\mathsfφ$ is a non-increasing service function $\mathsfφ : \mathbb{R}^+ \to \mathbb{R}^+$. This includes problems of placing $k$ base stations as to maximize the total bandwidth to the clients -- indeed, the closer the client is to its nearest base station, the more data it can send/receive, and the target is to place $k$ base stations so that the total bandwidth is maximized. We provide an $n^{\varepsilon^{-O(d)}}$ time algorithm for this problem that achieves a $(1-\varepsilon)$-approximation. Notably, the runtime does not depend on the parameter $k$ and it works for an arbitrary non-increasing service function $\mathsfφ : \mathbb{R}^+ \to \mathbb{R}^+$.
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