We study the classic \textsc{(Uncapacitated) Facility Location} problem on Unit Disk Graphs (UDGs). For a given point set $P$ in the plane, the unit disk graph UDG(P) on $P$ has vertex set $P$ and an edge between two distinct points $p, q \in P$ if and only if their Euclidean distance $|pq|$ is at most 1. The weight of the edge $pq$ is equal to their distance $|pq|$. An instance of \fl on UDG(P) consists of a set $C\subseteq P$ of clients and a set $F\subseteq P$ of facilities, each having an opening cost $f_i$. The goal is to pick a subset $F'\subseteq F$ to open while minimizing $\sum_{i\in F'} f_i + \sum_{v\in C} d(v,F')$, where $d(v,F')$ is the distance of $v$ to nearest facility in $F'$ through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

翻译：我们研究了单位圆盘图上的经典（无容量限制）设施选址问题。对于平面上给定的点集 $P$，其对应的单位圆盘图 UDG(P) 以 $P$ 为顶点集，且当两个不同点 $p, q \in P$ 的欧几里得距离 $|pq|$ 不超过 1 时，它们之间存在一条边。边 $pq$ 的权重等于其距离 $|pq|$。UDG(P) 上的设施选址问题实例由客户集 $C\subseteq P$ 和设施集 $F\subseteq P$ 组成，每个设施 $i$ 具有开放成本 $f_i$。目标是选取子集 $F'\subseteq F$ 进行开放，以最小化 $\sum_{i\in F'} f_i + \sum_{v\in C} d(v,F')$，其中 $d(v,F')$ 为 $v$ 通过 UDG(P) 到达 $F'$ 中最近设施的距离。本文提出了该问题的首个拟多项式时间近似方案。尽管稀疏几何图（如平面图）上的设施选址问题已有成熟的近似方案，但稠密图上的相关结果尚属空白。具体而言，据我们所知，在本研究之前，一般设置下的单位圆盘图上不存在任何设施选址问题的近似方案。