Topological stability of kinetic k-centers

Ivor van der Hoog, Marc van Kreveld, Wouter Meulemans, Kevin Verbeek, Jules Wulms

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

Abstract

We study the k-center problem in a kinetic setting: given a set of continuously moving points P in the plane, determine a set of k (moving) disks that cover P at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of k-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution—the topological stability ratio—considering various metrics and various optimization criteria. For k = 2 we provide tight bounds, and for small we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant k.

Original languageEnglish
Title of host publicationWALCOM
Subtitle of host publicationAlgorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings
EditorsShin-ichi Nakano, Gautam K. Das, Partha S. Mandal, Krishnendu Mukhopadhyaya
Place of PublicationCham
PublisherSpringer
Pages43-55
Number of pages13
ISBN (Electronic)978-3-030-10564-8
ISBN (Print)978-3-030-10563-1
DOIs
Publication statusPublished - 2019
Event13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019 - Guwahati, India
Duration: 27 Feb 20192 Mar 2019

Publication series

NameLecture Notes in Computer Science
Volume11355
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019
CountryIndia
CityGuwahati
Period27/02/192/03/19

Fingerprint

Kinetics
Upper and Lower Bounds
Radius
P-point
Center Problem
Polynomial time
Optimal Solution
Unstable
Polynomials
Cover
Metric
Optimization
Model

Keywords

  • Facility location
  • Stability analysis
  • Time-varying data

Cite this

van der Hoog, I., van Kreveld, M., Meulemans, W., Verbeek, K., & Wulms, J. (2019). Topological stability of kinetic k-centers. In S. Nakano, G. K. Das, P. S. Mandal, & K. Mukhopadhyaya (Eds.), WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings (pp. 43-55). (Lecture Notes in Computer Science; Vol. 11355). Cham: Springer. https://doi.org/10.1007/978-3-030-10564-8_4
van der Hoog, Ivor ; van Kreveld, Marc ; Meulemans, Wouter ; Verbeek, Kevin ; Wulms, Jules. / Topological stability of kinetic k-centers. WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings. editor / Shin-ichi Nakano ; Gautam K. Das ; Partha S. Mandal ; Krishnendu Mukhopadhyaya. Cham : Springer, 2019. pp. 43-55 (Lecture Notes in Computer Science).
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van der Hoog, I, van Kreveld, M, Meulemans, W, Verbeek, K & Wulms, J 2019, Topological stability of kinetic k-centers. in S Nakano, GK Das, PS Mandal & K Mukhopadhyaya (eds), WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings. Lecture Notes in Computer Science, vol. 11355, Springer, Cham, pp. 43-55, 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019, Guwahati, India, 27/02/19. https://doi.org/10.1007/978-3-030-10564-8_4

Topological stability of kinetic k-centers. / van der Hoog, Ivor; van Kreveld, Marc; Meulemans, Wouter; Verbeek, Kevin; Wulms, Jules.

WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings. ed. / Shin-ichi Nakano; Gautam K. Das; Partha S. Mandal; Krishnendu Mukhopadhyaya. Cham : Springer, 2019. p. 43-55 (Lecture Notes in Computer Science; Vol. 11355).

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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AB - We study the k-center problem in a kinetic setting: given a set of continuously moving points P in the plane, determine a set of k (moving) disks that cover P at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of k-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution—the topological stability ratio—considering various metrics and various optimization criteria. For k = 2 we provide tight bounds, and for small we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant k.

KW - Facility location

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KW - Time-varying data

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A2 - Mandal, Partha S.

A2 - Mukhopadhyaya, Krishnendu

PB - Springer

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van der Hoog I, van Kreveld M, Meulemans W, Verbeek K, Wulms J. Topological stability of kinetic k-centers. In Nakano S, Das GK, Mandal PS, Mukhopadhyaya K, editors, WALCOM: Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings. Cham: Springer. 2019. p. 43-55. (Lecture Notes in Computer Science). https://doi.org/10.1007/978-3-030-10564-8_4