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

International audienceWe 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 ε > 0, computes a solution of cost at most (1 + ε) times the optimum in time n 2 O(ε −2 log(1/ε))