### Abstract

We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center's capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80's. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7,8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

Original language | English |
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Title of host publication | Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 52-63 |

Number of pages | 12 |

ISBN (Print) | 9783319075563 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

Event | 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014 - Bonn, Germany Duration: 2014 Jun 23 → 2014 Jun 25 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8494 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014 |
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Country | Germany |

City | Bonn |

Period | 14/6/23 → 14/6/25 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings*(pp. 52-63). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8494 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-07557-0_5

}

*Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8494 LNCS, Springer Verlag, pp. 52-63, 17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014, Bonn, Germany, 14/6/23. https://doi.org/10.1007/978-3-319-07557-0_5

**Centrality of trees for capacitated k-center.** / An, Hyung-Chan; Bhaskara, Aditya; Chekuri, Chandra; Gupta, Shalmoli; Madan, Vivek; Svensson, Ola.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Centrality of trees for capacitated k-center

AU - An, Hyung-Chan

AU - Bhaskara, Aditya

AU - Chekuri, Chandra

AU - Gupta, Shalmoli

AU - Madan, Vivek

AU - Svensson, Ola

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center's capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80's. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7,8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

AB - We consider the capacitated k-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated non-negative integer capacity. The goal is to choose (open) k locations (called centers) and assign each location to an open center to minimize the maximum, over all locations, of the distance of the location to its assigned center. The number of locations assigned to a center cannot exceed the center's capacity. The uncapacitated k-center problem has a simple tight 2-approximation from the 80's. In contrast, the first constant factor approximation for the capacitated problem was obtained only recently by Cygan, Hajiaghayi and Khuller who gave an intricate LP-rounding algorithm that achieves an approximation guarantee in the hundreds. In this paper we give a simple algorithm with a clean analysis and prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7,8 or 9. The algorithm proceeds by first reducing to special tree instances, and then uses our best-possible algorithm to solve such instances. Our concept of tree instances is versatile and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are the same.

UR - http://www.scopus.com/inward/record.url?scp=84958540558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958540558&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-07557-0_5

DO - 10.1007/978-3-319-07557-0_5

M3 - Conference contribution

SN - 9783319075563

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 52

EP - 63

BT - Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings

PB - Springer Verlag

ER -