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

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 ⊆ V(G), a set of facilities F ⊆ V(G), and opening costs open: F → R_≥ 0, the goal is to find a subset D of F that minimizes ∑ _c∊C min_f∊D dist(c,f) + ∑ _f∊D 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 e>0, computes a solution of cost at most (1+e) times the optimum in time n^2^O(e^-2 log (1/e)).