Centrality of trees for capacitated k-center
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...
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01.12.2015
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Madan, Vivek Bhaskara, Aditya Gupta, Shalmoli An, Hyung-Chan Chekuri, Chandra Svensson, Ola |
| Author_xml | – sequence: 1 givenname: Hyung-Chan surname: An fullname: An, Hyung-Chan email: hyung-chan.an@epfl.ch organization: School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne – sequence: 2 givenname: Aditya surname: Bhaskara fullname: Bhaskara, Aditya organization: Google Research – sequence: 3 givenname: Chandra surname: Chekuri fullname: Chekuri, Chandra organization: University of Illinois at Urbana-Champaign – sequence: 4 givenname: Shalmoli surname: Gupta fullname: Gupta, Shalmoli organization: University of Illinois at Urbana-Champaign – sequence: 5 givenname: Vivek surname: Madan fullname: Madan, Vivek organization: University of Illinois at Urbana-Champaign – sequence: 6 givenname: Ola surname: Svensson fullname: Svensson, Ola organization: School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne |
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| DOI | 10.1007/s10107-014-0857-y |
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| Keywords | Approximation algorithms 68W25 LP-rounding algorithms Capacitated center problem Capacitated network location problems |
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| References | An, H.C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., Svensson, O.: Centrality of trees for capacitated k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center. In: IPCO, pp. 52–63 (2014) KorupoluMRPlaxtonCGRajaramanRAnalysis of a local search heuristic for facility location problemsJ. Algorithm.20003711461880962.68044178325210.1006/jagm.2000.1100 AryaVGargNKhandekarRMeyersonAMunagalaKPanditVLocal search heuristics for k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median and facility location problemsSIAM J. Comput.20043335445621105.68118206664110.1137/S0097539702416402 CharikarMGuhaSTardosÉShmoysDBA constant-factor approximation algorithm for the k-median problemJ. Comput. Syst. Sci.20026511291491023.90037194629110.1006/jcss.2002.1882 Chuzhoy, J., Rabani, Y.: Approximating k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median with non-uniform capacities. In: SODA, pp. 952–958 (2005) Bar-IlanJKortsarzGPelegDHow to allocate network centersJ. Algorithm.19931533854150784.68012123950110.1006/jagm.1993.1047 GonzalezTFClustering to minimize the maximum intercluster distanceTheor. Comput. Sci.1985382933060567.6204810.1016/0304-3975(85)90224-5 Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for k-centers with non-uniform hard capacities. In: FOCS, pp. 273–282 (2012) Levi, R., Shmoys, D.B., Swamy, C.: LP-based approximation algorithms for capacitated facility location. In: IPCO, pp. 206–218 (2004) Li, S., Svensson, O.: Approximating k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median problem via pseudo-approximation. In: STOC, pp. 901–910 (2013) KhullerSSussmannYJThe capacitated k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center problemSIAM J. Discret. Math.2000133403418178799110.1137/S0895480197329776 Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC, pp. 731–740 (2002) Shmoys, D.B., Tardos, É., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: STOC, pp. 265–274 (1997) HochbaumDSShmoysDBA best possible heuristic for the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center problemMath. Oper. Res.1985101801840565.9001579387610.1287/moor.10.2.180 ChudakFAWilliamsonDPImproved approximation algorithms for capacitated facility location problemsMath. Program.200510222072221079.90075212444310.1007/s10107-004-0524-9 Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: ICALP (2), pp. 77–88 (2011) ZhangJChenBYeYA multiexchange local search algorithm for the capacitated facility location problemMath. Oper. Res.20053023894031082.90057214203910.1287/moor.1040.0125 GuhaSKhullerSGreedy strikes back: improved facility location algorithmsJ. Algorithm.19993112282480928.68137168244010.1006/jagm.1998.0993 HallPOn representatives of subsetsJ. Lond. Math. Soc.1935102630 WilliamsonDPShmoysDBThe Design of Approximation Algorithms2011New YorkCambridge University Press1219.9000410.1017/CBO9780511921735 Bansal, M., Garg, N., Gupta, N.: A 5-approximation for capacitated facility location. In: ESA, pp. 133–144 (2012) Pál, M., Tardos, É., Wexler, T.: Facility location with nonuniform hard capacities. In: FOCS, pp. 329–338 (2001) JainKVaziraniVVApproximation algorithms for metric facility location and k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median problems using the primal-dual schema and lagrangian relaxationJ. ACM20014822742961138.90417186871710.1145/375827.375845 Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Approx-Random, pp. 29–43 (2007) CharikarMGuhaSImproved combinatorial algorithms for facility location problemsSIAM J. Comput.20053448038241075.68100214885910.1137/S0097539701398594 |
| References_xml | – reference: GonzalezTFClustering to minimize the maximum intercluster distanceTheor. Comput. Sci.1985382933060567.6204810.1016/0304-3975(85)90224-5 – reference: WilliamsonDPShmoysDBThe Design of Approximation Algorithms2011New YorkCambridge University Press1219.9000410.1017/CBO9780511921735 – reference: CharikarMGuhaSImproved combinatorial algorithms for facility location problemsSIAM J. Comput.20053448038241075.68100214885910.1137/S0097539701398594 – reference: ChudakFAWilliamsonDPImproved approximation algorithms for capacitated facility location problemsMath. Program.200510222072221079.90075212444310.1007/s10107-004-0524-9 – reference: HallPOn representatives of subsetsJ. Lond. Math. Soc.1935102630 – reference: CharikarMGuhaSTardosÉShmoysDBA constant-factor approximation algorithm for the k-median problemJ. Comput. Syst. Sci.20026511291491023.90037194629110.1006/jcss.2002.1882 – reference: Li, S., Svensson, O.: Approximating k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median problem via pseudo-approximation. In: STOC, pp. 901–910 (2013) – reference: Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: ICALP (2), pp. 77–88 (2011) – reference: HochbaumDSShmoysDBA best possible heuristic for the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center problemMath. Oper. Res.1985101801840565.9001579387610.1287/moor.10.2.180 – reference: JainKVaziraniVVApproximation algorithms for metric facility location and k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median problems using the primal-dual schema and lagrangian relaxationJ. ACM20014822742961138.90417186871710.1145/375827.375845 – reference: Levi, R., Shmoys, D.B., Swamy, C.: LP-based approximation algorithms for capacitated facility location. In: IPCO, pp. 206–218 (2004) – reference: KorupoluMRPlaxtonCGRajaramanRAnalysis of a local search heuristic for facility location problemsJ. Algorithm.20003711461880962.68044178325210.1006/jagm.2000.1100 – reference: Chuzhoy, J., Rabani, Y.: Approximating k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median with non-uniform capacities. In: SODA, pp. 952–958 (2005) – reference: Byrka, J.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. In: Approx-Random, pp. 29–43 (2007) – reference: AryaVGargNKhandekarRMeyersonAMunagalaKPanditVLocal search heuristics for k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-median and facility location problemsSIAM J. Comput.20043335445621105.68118206664110.1137/S0097539702416402 – reference: Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for k-centers with non-uniform hard capacities. In: FOCS, pp. 273–282 (2012) – reference: GuhaSKhullerSGreedy strikes back: improved facility location algorithmsJ. Algorithm.19993112282480928.68137168244010.1006/jagm.1998.0993 – reference: Bar-IlanJKortsarzGPelegDHow to allocate network centersJ. Algorithm.19931533854150784.68012123950110.1006/jagm.1993.1047 – reference: Pál, M., Tardos, É., Wexler, T.: Facility location with nonuniform hard capacities. In: FOCS, pp. 329–338 (2001) – reference: ZhangJChenBYeYA multiexchange local search algorithm for the capacitated facility location problemMath. Oper. Res.20053023894031082.90057214203910.1287/moor.1040.0125 – reference: An, H.C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., Svensson, O.: Centrality of trees for capacitated k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center. In: IPCO, pp. 52–63 (2014) – reference: Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC, pp. 731–740 (2002) – reference: Shmoys, D.B., Tardos, É., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: STOC, pp. 265–274 (1997) – reference: KhullerSSussmannYJThe capacitated k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-center problemSIAM J. Discret. Math.2000133403418178799110.1137/S0895480197329776 – reference: Bansal, M., Garg, N., Gupta, N.: A 5-approximation for capacitated facility location. In: ESA, pp. 133–144 (2012) |
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-center problem. In this problem we are given a finite set of locations in a metric space and each location has an associated... |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical |
| Title | Centrality of trees for capacitated k-center |
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