Feldmann, A.E. orcid.org/0000-0001-6229-5332 and Marx, D. (2020) The parameterized hardness of the k-center problem in transportation networks. Algorithmica, 82 (7). pp. 1989-2005. ISSN 0178-4617
Abstract
In this paper we study the hardness of the k-Center problem on inputs that model transportation networks. For the problem, a graph G= (V, E) with edge lengths and an integer k are given and a center set C⊆ V needs to be chosen such that | C| ≤ k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k-Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p. Moreover, under the exponential time hypothesis there is no f(k,p,h)·no(p+k+h) time algorithm for any computable function f. Thus it is unlikely that the optimum solution to k-Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1 + ε) -approximation algorithm for inputs of doubling dimension d with runtime (kk/ εO(kd)) · nO(1). This generalizes a previous result, which considered inputs in D-dimensional Lq metrics.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 Springer Science+Business Media, LLC, part of Springer Nature. This is an author-produced version of a paper subsequently published in Algorithmica. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | k-Center problem; Parameterized complexity; Planar graphs; Highway dimension; Doubling dimension; Pathwidth |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 27 Jun 2023 15:20 |
Last Modified: | 28 Jun 2023 09:33 |
Status: | Published |
Publisher: | Springer Science and Business Media LLC |
Refereed: | Yes |
Identification Number: | 10.1007/s00453-020-00683-w |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:200953 |