Submodular Clustering in Low Dimensions

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 $\sum_{p \in P}\mathsf{\varphi}\bigl( \mathsf{d}(p,C) \bigr)$ to the points $P$, where $\mathsf{d}(p,C)$ is the distance of $p$ to its nearest center in $C$, and $\mathsf{\varphi}$ is a non-increasing service function $\mathsf{\varphi} : \mathbb{R}^+ \to \mathbb{R}^+$. 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 $\mathsf{\varphi} : \mathbb{R}^+ \to \mathbb{R}^+$.