Continuous facility location on graphs

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

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

Continuous facility location on graphs

Tim A. Hartmann, Stefan Lendl, Gerhard J. Woeginger^*

^*Corresponding author for this work

Institute of Discrete Mathematics (5050)

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Abstract

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned at the vertices as well as at interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range δ > 0. In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most δ from one of these facilities. We investigate this covering problem in terms of the rational parameter δ. We prove that the problem is polynomially solvable whenever δ is a unit fraction, and that the problem is NP-hard for all non unit fractions δ. We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for all δ < 3/2, and it is W[2]-hard for all δ ≥ 3/2.

Original language	English
Title of host publication	Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings
Subtitle of host publication	21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings
Editors	Daniel Bienstock, Giacomo Zambelli
Place of Publication	Cham
Publisher	Springer
Pages	171-181
Number of pages	11
ISBN (Print)	9783030457709
DOIs	https://doi.org/10.1007/978-3-030-45771-6_14
Publication status	Published - 1 Jan 2020
Event	21st Conference on Integer Programming and Combinatorial Optimization: IPCO 2020 - London School of Economics, Virtuell, United Kingdom Duration: 8 Jun 2020 → 10 Jun 2020 Conference number: 21 http://www.lse.ac.uk/IPCO-2020

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12125 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	21st Conference on Integer Programming and Combinatorial Optimization
Abbreviated title	IPCO 2020 LSE
Country/Territory	United Kingdom
City	Virtuell
Period	8/06/20 → 10/06/20
Internet address	http://www.lse.ac.uk/IPCO-2020

Keywords

Graph theory
Location theory
Parametrized complexity

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

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

Cite this

Hartmann, T. A., Lendl, S., & Woeginger, G. J. (2020). Continuous facility location on graphs. In D. Bienstock, & G. Zambelli (Eds.), Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings (pp. 171-181). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12125 LNCS). Springer. https://doi.org/10.1007/978-3-030-45771-6_14

Continuous facility location on graphs. / Hartmann, Tim A.; Lendl, Stefan; Woeginger, Gerhard J.
Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings. ed. / Daniel Bienstock; Giacomo Zambelli. Cham: Springer, 2020. p. 171-181 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12125 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Hartmann, TA, Lendl, S & Woeginger, GJ 2020, Continuous facility location on graphs. in D Bienstock & G Zambelli (eds), Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12125 LNCS, Springer, Cham, pp. 171-181, 21st Conference on Integer Programming and Combinatorial Optimization, Virtuell, United Kingdom, 8/06/20. https://doi.org/10.1007/978-3-030-45771-6_14

Hartmann TA, Lendl S, Woeginger GJ. Continuous facility location on graphs. In Bienstock D, Zambelli G, editors, Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings. Cham: Springer. 2020. p. 171-181. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-030-45771-6_14

Hartmann, Tim A. ; Lendl, Stefan ; Woeginger, Gerhard J. / Continuous facility location on graphs. Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings. editor / Daniel Bienstock ; Giacomo Zambelli. Cham : Springer, 2020. pp. 171-181 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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N2 - We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned at the vertices as well as at interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range δ > 0. In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most δ from one of these facilities. We investigate this covering problem in terms of the rational parameter δ. We prove that the problem is polynomially solvable whenever δ is a unit fraction, and that the problem is NP-hard for all non unit fractions δ. We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for all δ < 3/2, and it is W[2]-hard for all δ ≥ 3/2.

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Continuous facility location on graphs

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Keywords

ASJC Scopus subject areas

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