A Polynomial-Time Approximation Scheme for Facility Location on Planar Graphs

We consider the classic Facility Location problem on planar graphs (non-uniform, uncapacitated). Given an edge-weighted planar graph $G$, a set of clients $C\subseteq V(G)$, a set of facilities $F\subseteq V(G)$, and opening costs $\mathsf{open} \colon F \to \mathbb{R}{\geq 0}$, the goal is to find a subset $D$ of $F$ that minimizes $\sum{c \in C} \min{f \in D} \mathrm{dist}(c,f) + \sum{f \in D} \mathsf{open}(f)$. The Facility Location problem remains one of the most classic and fundamental optimization problem for which it is not known whether it admits a polynomial-time approximation scheme (PTAS) on planar graphs despite significant effort for obtaining one. We solve this open problem by giving an algorithm that for any $\varepsilon>0$, computes a solution of cost at most $(1+\varepsilon)$ times the optimum in time $n^{{2{O(\varepsilon{-2}} \log (1/\varepsilon))}}$.