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 $\mathsf{o}\mathsf{p}\mathsf{e}\mathsf{n}\text{\hspace{0.17em}⁣}:F\to {\mathbb{R}}_{\ge 0}$, the goal is to find a subset $D$ of $F$ that minimizes $\sum_{c \in C} \min_{f \in D} $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}$(c,f) + \sum_{f \in D} $\mathsf{o}\mathsf{p}\mathsf{e}\mathsf{n}$(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 $\epsilon >0$, computes a solution of cost at most $(1+\epsilon )$ times the optimum in time $n^{2^{O(\varepsilon^{-2} \log (1/\varepsilon))}}$.