Centrality of trees for capacitated k-center

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

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We consider the capacitated $$k$$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) $$k$$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 $$k$$k-center problem has a simple tight (Formula presented.)-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 (Formula presented.). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either (Formula presented.). 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 $$k$$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
Pages (from-to)29-53
Number of pages25
JournalMathematical Programming
Volume154
Issue number1-2
DOIs
Publication statusPublished - 2015 Dec 1

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

  • Software
  • Mathematics(all)

Cite this

An, H. C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., & Svensson, O. (2015). Centrality of trees for capacitated k-center. Mathematical Programming, 154(1-2), 29-53. https://doi.org/10.1007/s10107-014-0857-y
An, Hyung Chan ; Bhaskara, Aditya ; Chekuri, Chandra ; Gupta, Shalmoli ; Madan, Vivek ; Svensson, Ola. / Centrality of trees for capacitated k-center. In: Mathematical Programming. 2015 ; Vol. 154, No. 1-2. pp. 29-53.
@article{23cc6d51dfc645d48735d2ac8c34d207,
title = "Centrality of trees for capacitated k-center",
abstract = "We consider the capacitated $$k$$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) $$k$$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 $$k$$k-center problem has a simple tight (Formula presented.)-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 (Formula presented.). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either (Formula presented.). 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 $$k$$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.",
author = "An, {Hyung Chan} and Aditya Bhaskara and Chandra Chekuri and Shalmoli Gupta and Vivek Madan and Ola Svensson",
year = "2015",
month = "12",
day = "1",
doi = "10.1007/s10107-014-0857-y",
language = "English",
volume = "154",
pages = "29--53",
journal = "Mathematical Programming",
issn = "0025-5610",
publisher = "Springer-Verlag GmbH and Co. KG",
number = "1-2",

}

An, HC, Bhaskara, A, Chekuri, C, Gupta, S, Madan, V & Svensson, O 2015, 'Centrality of trees for capacitated k-center', Mathematical Programming, vol. 154, no. 1-2, pp. 29-53. https://doi.org/10.1007/s10107-014-0857-y

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

In: Mathematical Programming, Vol. 154, No. 1-2, 01.12.2015, p. 29-53.

Research output: Contribution to journalArticle

TY - JOUR

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 - 2015/12/1

Y1 - 2015/12/1

N2 - We consider the capacitated $$k$$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) $$k$$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 $$k$$k-center problem has a simple tight (Formula presented.)-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 (Formula presented.). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either (Formula presented.). 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 $$k$$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 $$k$$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) $$k$$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 $$k$$k-center problem has a simple tight (Formula presented.)-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 (Formula presented.). It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either (Formula presented.). 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 $$k$$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=84946400386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946400386&partnerID=8YFLogxK

U2 - 10.1007/s10107-014-0857-y

DO - 10.1007/s10107-014-0857-y

M3 - Article

VL - 154

SP - 29

EP - 53

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -

An HC, Bhaskara A, Chekuri C, Gupta S, Madan V, Svensson O. Centrality of trees for capacitated k-center. Mathematical Programming. 2015 Dec 1;154(1-2):29-53. https://doi.org/10.1007/s10107-014-0857-y