A PTAS for the Geometric Connected Facility Location Problem

Abstract

We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients \(\mathcal {C} \subset \mathbb {R}^{d}\), one wants to select a set of locations \(F \subset \mathbb {R}^{d}\) where to open facilities, each at a fixed cost f≥0. For each client \(j \in \mathcal {C}\), 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 (T), where M≥1 and (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.

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Notes

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    That is, the vertex \((j,i^{\prime }) \in \mathcal {V}\) that represents the connected component \(i^{\prime }\) of G j is part of connected component i of G.

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Correspondence to Rafael C. S. Schouery.

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This work was partially supported by grants 2013/21744-8 , 2014/14209-1 and 2013/21744-8, São Paulo Research Foundation (FAPESP) and grants 311499/2014-7 and 477692/2012-5, National Council for Scientific and Technological Development (CNPq).

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Miyazawa, F.K., C. Pedrosa, L.L., S. Schouery, R.C. et al. A PTAS for the Geometric Connected Facility Location Problem. Theory Comput Syst 61, 871–892 (2017). https://doi.org/10.1007/s00224-017-9749-x

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Keywords

  • Connected facility location problem
  • Geometric problem
  • Polynomial-time approximation scheme
  • Prize-collecting