Centrality of trees for capacitated k-center

Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, Ola Svensson

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Citations (Scopus)

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 languageEnglish
Title of host publicationInteger Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings
PublisherSpringer Verlag
Pages52-63
Number of pages12
ISBN (Print)9783319075563
DOIs
Publication statusPublished - 2014 Jan 1
Event17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014 - Bonn, Germany
Duration: 2014 Jun 232014 Jun 25

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8494 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other17th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2014
CountryGermany
CityBonn
Period14/6/2314/6/25

Fingerprint

Centrality
Center Problem
Integrality
Approximation
Preprocessing
LP Rounding
LP Relaxation
Approximation algorithms
Beat
Metric space
Assign
Approximation Algorithms
Finite Set
Exceed
Choose
Non-negative
Minimise
Integer
Necessary

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

An, H-C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., & Svensson, O. (2014). Centrality of trees for capacitated k-center. In 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
An, Hyung-Chan ; Bhaskara, Aditya ; Chekuri, Chandra ; Gupta, Shalmoli ; Madan, Vivek ; Svensson, Ola. / Centrality of trees for capacitated k-center. Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings. Springer Verlag, 2014. pp. 52-63 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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An, H-C, Bhaskara, A, Chekuri, C, Gupta, S, Madan, V & Svensson, O 2014, Centrality of trees for capacitated k-center. in 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.

Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings. Springer Verlag, 2014. p. 52-63 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8494 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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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.

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An H-C, Bhaskara A, Chekuri C, Gupta S, Madan V, Svensson O. Centrality of trees for capacitated k-center. In Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Proceedings. Springer Verlag. 2014. p. 52-63. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-07557-0_5