A PTAS for the Geometric Connected Facility Location Problem

We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients C⊂ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C} \subset \mathbb {R}^{d}$\end{document}, one wants to select a set of locations F⊂ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F \subset \mathbb {R}^{d}$\end{document} where to open facilities, each at a fixed cost f≥0. For each client j∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$j \in \mathcal {C}$\end{document}, one has to choose to either connect it to an open facility ϕ(j)∈F paying the Euclidean distance between j and ϕ(j), or pay a given penalty cost π(j). The facilities must also be connected by a tree T, whose cost is Mℓ(T), where M≥1 and ℓ(T) is the total Euclidean length of edges in T. The multiplication by M reflects the fact that interconnecting two facilities is typically more expensive than connecting a client to a facility. The objective is to find a solution with minimum cost. In this paper, we present a Polynomial-Time Approximation Scheme (PTAS) for the two-dimensional GCFLP. Our algorithm also leads to a PTAS for the two-dimensional Geometric Connected k-medians, when f=0, but only k facilities may be opened.