### 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 language | English |
---|---|

Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 13th International Conference, WALCOM 2019, Proceedings |

Editors | Shin-ichi Nakano, Gautam K. Das, Partha S. Mandal, Krishnendu Mukhopadhyaya |

Place of Publication | Cham |

Publisher | Springer |

Pages | 43-55 |

Number of pages | 13 |

ISBN (Electronic) | 978-3-030-10564-8 |

ISBN (Print) | 978-3-030-10563-1 |

DOIs | |

Publication status | Published - 2019 |

Event | 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019 - Guwahati, India Duration: 27 Feb 2019 → 2 Mar 2019 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Volume | 11355 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Conference and Workshop on Algorithms and Computations, WALCOM 2019 |
---|---|

Country | India |

City | Guwahati |

Period | 27/02/19 → 2/03/19 |

### Fingerprint

### Keywords

- Facility location
- Stability analysis
- Time-varying data

### Cite this

*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

}

*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.

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

TY - GEN

T1 - Topological stability of kinetic k-centers

AU - van der Hoog, Ivor

AU - van Kreveld, Marc

AU - Meulemans, Wouter

AU - Verbeek, Kevin

AU - Wulms, Jules

PY - 2019

Y1 - 2019

N2 - 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.

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

KW - Stability analysis

KW - Time-varying data

UR - http://www.scopus.com/inward/record.url?scp=85062680763&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-10564-8_4

DO - 10.1007/978-3-030-10564-8_4

M3 - Conference contribution

AN - SCOPUS:85062680763

SN - 978-3-030-10563-1

T3 - Lecture Notes in Computer Science

SP - 43

EP - 55

BT - WALCOM

A2 - Nakano, Shin-ichi

A2 - Das, Gautam K.

A2 - Mandal, Partha S.

A2 - Mukhopadhyaya, Krishnendu

PB - Springer

CY - Cham

ER -