Continuous Facility Location on Graphs

Hartmann, Tim A.; Lendl, Stefan; Woeginger, Gerhard J.

doi:10.1007/978-3-030-45771-6_14

Cited by 2 publications

(5 citation statements)

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“…In other words, the objective in dcovering is not to pack the facilities, but to cover the set P(G) with them. Hartmann, Lendl and Woeginger [13] have shown that d-covering is easy if the numerator of the rational number d is 1, wheras all other cases with rational d turn out to be NPcomplete.…”

Section: Discussionmentioning

confidence: 99%

Dispersing Obnoxious Facilities on a Graph

et al. 2021

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We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

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Section: Discussionmentioning

confidence: 99%

Dispersing Obnoxious Facilities on a Graph

et al. 2021

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“…where, if e ∈ L, the term τ ve l i (q e ) in (6n) is replaced by d(v, v i ) + 1 i=a q e + 1 i=b (2δu e + le − q e ). We enforce (17) because e always contains a facility if e ∈ L. On the other hand, we need to include constraints (18) to guarantee that the variables r va and r v b can take positive values. Indeed, if w e = 1 for e ∈ L, it may happen that x va = 1 or x v b = 1, which will will imply, respectively, r va = 0 or r v b = 0 due to (6l).…”

Section: Network Processingmentioning

confidence: 99%

“…However, this assumption corresponds to ideal but usually unrealistic scenarios (the reader is referred to the examples of real applications of the above paragraph). Some works addressing network covering with continuous sets of both candidate locations and demand points are [12,13,14], for maximal covering, and [15,16,17], for set-covering. We focus on the latter variant, which we call the continuous set-covering problem.…”

Section: Introductionmentioning

confidence: 99%

“…The authors presented three different approaches to solve the problem, including a Mixed Integer Linear Programming (MILP) formulation. On the other hand, Hartmann et al [17] focused on the computational complexity of the continuous set-covering for general covering radii. They proved that, when all edges have unit length, the continuous set-covering is polynomially solvable if the covering radius is a unit fraction, and is NP-hard otherwise.…”

Section: Introductionmentioning

confidence: 99%

“…The CSCP δ is known to be NP-hard, see [17]. A first observation is that typical simplifications proposed for other studied variants of set-covering are not valid for the CSCP δ .…”

Section: Introductionmentioning

confidence: 99%

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Continuous Covering on Networks: Improved Mixed Integer Programming Formulations

Xu¹

2022

Preprint

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Covering problems are well-studied in the domain of Operations Research, and, more specifically, in Location Science. When the location space is a network, the most frequent assumption is to consider the candidate facility locations, the points to be covered, or both, to be discrete sets. In this work, we study the set-covering location problem when both candidate locations and demand points are continuous sets on a network. This variant has received little attention, and the scarce existing approaches have focused on particular cases, such as tree networks and integer covering radius. Here we study the general problem and present a Mixed Integer Linear Programming formulation (MILP) for networks with edges' lengths no greater than the covering radius. The model does not lose generality, as any edge not satisfying this condition can be partitioned into subedges of appropriate lengths without changing the problem. We propose a preprocessing algorithm to reduce the size of the MILP, and devise tight big-M constants and valid inequalities to strengthen our formulations.Moreover, a second MILP is proposed, which admits edges' lengths greater than the covering radius. As opposed to existing formulations of the problem (including the first MILP proposed herein), the number of variables and constraints of this second model does not depend on the lengths of the network's edges. This second model represents a scalable approach that particularly suits real-world networks, whose edges are usually greater than the covering radius. Our computational experiments show the strengths and limitations of our exact approach on both real-world and random networks. Our formulations are also tested against an existing exact method.

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Continuous Facility Location on Graphs

Cited by 2 publications

References 5 publications

Dispersing Obnoxious Facilities on a Graph

Dispersing Obnoxious Facilities on a Graph

Continuous Covering on Networks: Improved Mixed Integer Programming Formulations

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