  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.
