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

04/24/2019

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by   Vincent Cohen-Addad, et al.

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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_f ∈ Ddist(c,f) + ∑_f ∈ Dopen(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 (1/ε)).