A PTAS for the Geometric Connected Facility Location Problem

A PTAS for the Geometric Connected Facility Location Problem We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients C ⊂ ℝ d $\mathcal {C} \subset \mathbb {R}^{d}$ , one wants to select a set of locations F ⊂ ℝ d $F \subset \mathbb {R}^{d}$ where to open facilities, each at a fixed cost f≥0. For each client j ∈ C $j \in \mathcal {C}$ , 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 M ℓ(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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

A PTAS for the Geometric Connected Facility Location Problem

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-017-9749-x
Publisher site
See Article on Publisher Site

Abstract

We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients C ⊂ ℝ d $\mathcal {C} \subset \mathbb {R}^{d}$ , one wants to select a set of locations F ⊂ ℝ d $F \subset \mathbb {R}^{d}$ where to open facilities, each at a fixed cost f≥0. For each client j ∈ C $j \in \mathcal {C}$ , 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 M ℓ(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.

Journal

Theory of Computing SystemsSpringer Journals

Published: Feb 2, 2017

References

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