## A constant-factor approximation algorithm for the k-median problem (1999)

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Venue: | In Proceedings of the 31st Annual ACM Symposium on Theory of Computing |

Citations: | 212 - 13 self |

### BibTeX

@INPROCEEDINGS{Charikar99aconstant-factor,

author = {Moses Charikar and Sudipto Guha and Éva Tardos and David B. Shmoys},

title = {A constant-factor approximation algorithm for the k-median problem},

booktitle = {In Proceedings of the 31st Annual ACM Symposium on Theory of Computing},

year = {1999},

pages = {1--10}

}

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### Abstract

We present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric k-median problem, we are given n points in a metric space. We select k of these to be cluster centers, and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 6 2 3-approximation algorithm for this problem. This improves upon the best previously known result of O(log k log log k), which was obtained by refining and derandomizing a randomized O(log n log log n)-approximation algorithm of Bartal. 1

### Citations

319 | Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation
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(Show Context)
Citation Context ... of cost at most 16 times the true optimal cost, but allows centers to be assigned at most 4U locations. There has also been a great deal of subsequent work to improve on our results. Jain & Vazirani =-=[20]-=- give an extremely elegant primal-dual 3-approximation algorithm for the uncapacitated facility location problem, and show how to use that procedure to obtain a 6-approximation algorithm for the k-med... |

314 | Probabilistic approximation of metric spaces and its algorithmic applications
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Citation Context ...mes the optimum, while using at most (1+1/ɛ)k cluster centers. The first non-trivial approximation algorithm that produces a feasible solution (i.e., one that uses at most k centers) is due to Bartal =-=[4, 5]-=-. By combining his result on the approximation of any metric by tree metrics with the fact that the k-median problem can be solved optimally in a tree metric, Bartal gave a randomized O(log n log log ... |

281 |
Clustering to minimize the maximum intercluster distance
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Citation Context ...s [18] and subsequently Dyer & Frieze [13] gave 2-approximation algorithms for the metric case problem (which is best possible unless P = NP), and also gave extensions for weighted variants. Gonzalez =-=[15]-=- considered the variant in which the objective is to minimize maximum distance between a pair of points in the same cluster, and independently gave a 2-approximation algorithm (which is also best poss... |

261 | Approximation algorithms for facility location problems
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- 1997
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Citation Context ...nment cost of each location to its closest open facility, where we assume that the cost of assigning location j to facility i is proportional to the distance between i and j. Shmoys, Tardos, & Aardal =-=[28]-=- use the techniques of Lin & Vitter [24] to give the first constant-factor approximation algorithm for the metric uncapacitated facility location problem. The quality of approximation has been improve... |

253 | On approximating arbitrary metrices by tree metrics
- Bartal
- 1998
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Citation Context ...mes the optimum, while using at most (1+1/ɛ)k cluster centers. The first non-trivial approximation algorithm that produces a feasible solution (i.e., one that uses at most k centers) is due to Bartal =-=[4, 5]-=-. By combining his result on the approximation of any metric by tree metrics with the fact that the k-median problem can be solved optimally in a tree metric, Bartal gave a randomized O(log n log log ... |

229 | Local search heuristic for k-median and facility location problems
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Citation Context ...r every k>0, the first k locations in this order serve as a k-median solution of cost that is within a constant factor of optimal. Most recently, Arya, Garg, Khandekar, Meyerson, Munagala, and Pandit =-=[2]-=- showed that the natural local search heuristic that interchanges p locations as centers/non-centers is guaranteed to find a solution that is within a factor of 3 + (2/p), for any positive integer p. ... |

206 |
An Algorithmic Approach to Network Location Problems II: The p-medians
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- 1979
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Citation Context ...supported by NSF grants CCR-9912422 and DMS-9505155 and ONR grant N00014-96-1-00500. 1sConsequently, it is quite natural to consider special cases. The problem is solvable in polynomial time on trees =-=[21, 30]-=-. However, for general metric spaces, the problem is NP-hard to solve exactly. Arora, Raghavan & Rao [1] give a polynomial-time approximation scheme for the k-median problem with 2-dimensional Euclide... |

205 | Improved combinatorial algorithms for facility location problems
- Charikar, Guha
(Show Context)
Citation Context ...imal-dual 3-approximation algorithm for the uncapacitated facility location problem, and show how to use that procedure to obtain a 6-approximation algorithm for the k-median problem. Charikar & Guha =-=[8]-=- refine this result to obtain a 4-approximation algorithm for the k-median problem, in addition to providing a number of new techniques that, combined with the previously known algorithms, yield a 1.7... |

183 | Greedy strikes back: improved facility location algorithms
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- 1998
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Citation Context ... & Vitter [24] to give the first constant-factor approximation algorithm for the metric uncapacitated facility location problem. The quality of approximation has been improved in a sequence of papers =-=[17, 9, 10]-=-. The best result previously known is a (1 + 2/e)-approximation algorithm due to Chudak & Shmoys [9, 10] (though, as we shall briefly discuss below, subsequent to the proceedings version of this paper... |

153 | An approximation algorithm for the generalized assignment problem - Shmoys, Tardos - 1993 |

146 | Analysis of a local search heuristic for facility location problems
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Citation Context ...the assumption that P�= NP. All of these algorithms rely on solving a natural linear programming relaxation of the problem and rounding the optimal fractional solution. Korupolu, Plaxton, & Rajaraman =-=[23]-=- analyze variants of simple local search algorithms for several clustering problems and show, for example, that for any ɛ>0, this leads to a (5 + ɛ)-approximation algorithm for the uncapacitated facil... |

122 | Improved approximation algorithms for uncapaciateted facility location
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- 1998
(Show Context)
Citation Context ... & Vitter [24] to give the first constant-factor approximation algorithm for the metric uncapacitated facility location problem. The quality of approximation has been improved in a sequence of papers =-=[17, 9, 10]-=-. The best result previously known is a (1 + 2/e)-approximation algorithm due to Chudak & Shmoys [9, 10] (though, as we shall briefly discuss below, subsequent to the proceedings version of this paper... |

114 | Approximation schemes for euclidean k-medians and related problems
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- 1998
(Show Context)
Citation Context ...ite natural to consider special cases. The problem is solvable in polynomial time on trees [21, 30]. However, for general metric spaces, the problem is NP-hard to solve exactly. Arora, Raghavan & Rao =-=[1]-=- give a polynomial-time approximation scheme for the k-median problem with 2-dimensional Euclidean inputs. We study the metric k-median problem, that is, we assume that the input points are located in... |

91 |
The Uncapacitated Facility Location Problem
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- 1990
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Citation Context ...n problem is closely related to the uncapacitated facility location problem, which is a central problem in the Operations Research literature (see, e.g., the survey of Cornuéjols, Nemhauser, & Wolsey =-=[12]-=-). In this problem, each location has a cost fi, the cost of opening a center (or facility) at location i. There is no restriction on the number of facilities that can be opened, but instead the goal ... |

78 | Improved approximation algorithms for capacitated facility location problems - Chudak, Williamson - 2005 |

77 |
epsilon-approximations with minimum packing constraint violation
- Lin, Vitter
- 1992
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Citation Context ...a 6 2-approximation algorithm for this problem, that is, a polynomial-time algorithm that 3 finds a feasible solution of objective function value within a factor of 6 2 3 of the optimum. Lin & Vitter =-=[25]-=- considered the k-median problem with arbitrary assignment costs, and gave a polynomial-time algorithm that finds, for any ɛ>0, a solution for which the objective function value is within a factor of ... |

69 | Approximation algorithms for geometric median problems
- Lin, Vitter
- 1992
(Show Context)
Citation Context ...an problem, that is, we assume that the input points are located in a metric space, or in other words, assume that the assignment costs are symmetric and satisfy the triangle inequality. Lin & Vitter =-=[24]-=- also gave a polynomial-time algorithm for the metric k-median problem that, for any ɛ>0, finds a solution of cost no more than 2(1+ɛ) times the optimum, while using at most (1+1/ɛ)k cluster centers. ... |

64 | How to allocate network centers - Bar-Ilan, Kortsarz, et al. - 1993 |

55 | Rounding via trees: Deterministic approximation algorithms for group Steiner trees and k-median
- Charikar, Chekuri, et al.
- 1998
(Show Context)
Citation Context ...n)-approximation algorithm for the k-median problem. This algorithm was subsequently derandomized and refined to yield an O(log k log log k)-approximation algorithm by Charikar, Chekuri, Goel, & Guha =-=[7]-=-. Approximation algorithms have been studied for a variety of clustering problems. The k-center problem is the min-max analogue of the k-median problem: one opens centers at k locations out of n, so a... |

49 |
An O(pn 2 ) algorithm for the p-median and related problems on tree graphs
- Tamir
- 1996
(Show Context)
Citation Context ...supported by NSF grants CCR-9912422 and DMS-9505155 and ONR grant N00014-96-1-00500. 1sConsequently, it is quite natural to consider special cases. The problem is solvable in polynomial time on trees =-=[21, 30]-=-. However, for general metric spaces, the problem is NP-hard to solve exactly. Arora, Raghavan & Rao [1] give a polynomial-time approximation scheme for the k-median problem with 2-dimensional Euclide... |

37 |
A simple heuristic for the p-center problem
- Dyer, Frieze
- 1985
(Show Context)
Citation Context ...roblem: one opens centers at k locations out of n, so as to minimize the maximum distance that an unselected location is from its nearest center. Hochbaum & Shmoys [18] and subsequently Dyer & Frieze =-=[13]-=- gave 2-approximation algorithms for the metric case problem (which is best possible unless P = NP), and also gave extensions for weighted variants. Gonzalez [15] considered the variant in which the o... |

35 |
A best possible approximation algorithm for the k–Center problem
- Hochbaum, Shmoys
- 1985
(Show Context)
Citation Context ...e min-max analogue of the k-median problem: one opens centers at k locations out of n, so as to minimize the maximum distance that an unselected location is from its nearest center. Hochbaum & Shmoys =-=[18]-=- and subsequently Dyer & Frieze [13] gave 2-approximation algorithms for the metric case problem (which is best possible unless P = NP), and also gave extensions for weighted variants. Gonzalez [15] c... |

33 | The capacitated k-center problem - Khuller, Sussmann - 1996 |

32 | approximations with minimum packing constraint violation - Lin, Vitter - 1992 |

19 | An O(pn) algorithm for p-median and related problems on tree graphs - Tamir - 1996 |

13 |
ImprovedCombinatorial Algorithms for Facility Location Problems
- Charikar, Guha
- 2005
(Show Context)
Citation Context ...ntee for the variant of the k-median problem where the objective is to minimize the sum of the squares of the distances from the vertices to their nearest centers. More substantially, Charikar & Guha =-=[6, 16]-=- consider a capacitated version of the k-median problem, in which each center can be assigned at most U locations. Charikar & Guha give a polynomial-time algorithm that finds a solution of cost at mos... |

11 |
The online median problem
- Mettu, Plaxton
- 2000
(Show Context)
Citation Context ...r of new techniques that, combined with the previously known algorithms, yield a 1.728-approximation algorithm for the uncapacitated facility location problem. For the median problem, Mettu & Plaxton =-=[26]-=- give the quite surprising result that one can efficiently compute a single permutation of the locations such that, for every k>0, the first k locations in this order serve as a k-median solution of c... |

3 |
Properties of the tree K-median linear programming relaxation. Unpublished manuscript
- Ward, Wong, et al.
- 1994
(Show Context)
Citation Context ...n. The linear programming relaxation is analogous to the relaxation used by the approximation algorithms for the facility location problem, and has also been studied for the k-median problem on trees =-=[32, 31]-=-. 2sWe obtain our result by combining the filtering technique of Lin & Vitter [25] with a more sophisticated method for selecting which centers to open. The filtering technique of Lin & Vitter [25] gu... |

2 |
The Kmedian Problem on a Tree. Working paper
- Vries, Posner, et al.
- 1998
(Show Context)
Citation Context ...n. The linear programming relaxation is analogous to the relaxation used by the approximation algorithms for the facility location problem, and has also been studied for the k-median problem on trees =-=[32, 31]-=-. 2sWe obtain our result by combining the filtering technique of Lin & Vitter [25] with a more sophisticated method for selecting which centers to open. The filtering technique of Lin & Vitter [25] gu... |

1 |
Algorithms for Clustering Problems
- Charikar
- 2000
(Show Context)
Citation Context ...ntee for the variant of the k-median problem where the objective is to minimize the sum of the squares of the distances from the vertices to their nearest centers. More substantially, Charikar & Guha =-=[6, 16]-=- consider a capacitated version of the k-median problem, in which each center can be assigned at most U locations. Charikar & Guha give a polynomial-time algorithm that finds a solution of cost at mos... |

1 |
A new greedy approach for facililty location problems
- Jain, Mahdian, et al.
- 2002
(Show Context)
Citation Context ...ic that interchanges p locations as centers/non-centers is guaranteed to find a solution that is within a factor of 3 + (2/p), for any positive integer p. On the other hand, Jain, Mahdian, and Saberi =-=[19]-=- have shown that no performance guarantee better than 1 + (2/e) can be obtained in polynomial time, unless NP ⊆ DTIME[n O(log log n) ]. 2 The k-median problem It is more natural to state our algorithm... |