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

## TL;DR

This paper presents a polynomial-time approximation scheme for the Facility Location problem on planar graphs, achieving near-optimal solutions efficiently and resolving a long-standing open problem in the field.

## Contribution

The authors develop the first PTAS for Facility Location on planar graphs, providing a significant advancement in approximation algorithms for this classic problem.

## Key findings

- Achieves a (1+ε)-approximate solution in quasi-polynomial time.
- Resolves the open problem of PTAS existence for Facility Location on planar graphs.
- Provides a new algorithmic framework for similar geometric and graph problems.

## Abstract

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 $n^{2^{O(\varepsilon^{-2} \log (1/\varepsilon))}}$.