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 ⊆ 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 Σ cϵC min fϵD dist(c,f) + Σ fϵD 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)) ) .