## Facility location in sublinear time (2005)

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Venue: | In 32nd International Colloquium on Automata, Languages, and Programming |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Bădoiu05facilitylocation,

author = {Mihai Bădoiu and Artur Czumaj and Piotr Indyk and Christian Sohler},

title = {Facility location in sublinear time},

booktitle = {In 32nd International Colloquium on Automata, Languages, and Programming},

year = {2005},

pages = {866--877}

}

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### Abstract

Abstract. In this paper we present a randomized constant factor approximation algorithm for the problem of computing the optimal cost of the metric Minimum Facility Location problem, in the case of uniform costs and uniform demands, and in which every point can open a facility. By exploiting the fact that we are approximating the optimal cost without computing an actual solution, we give the first algorithm for this problem with running time O(n log 2 n), where n is the number of metric space points. Since the size of the representation of an n-point metric space is Θ(n 2), the complexity of our algorithm is sublinear with respect to the input size. We consider also the general version of the metric Minimum Facility Location problem and we show that there is no o(n 2)-time algorithm, even a randomized one, that approximates the optimal solution to within any factor. This result can be generalized to some related problems, and in particular, the cost of minimum-cost matching, the cost of bichromatic matching, or the cost of n/2-median cannot be approximated in o(n 2)-time. 1

### Citations

322 | Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation
- Jain, Vazirani
(Show Context)
Citation Context ...ation ratio is due to Guha and Khuller [5]. The first constant factor approximation algorithm with almost linear running time (that is, the running time of O(n2 log n)) was given by Jain and Vazirani =-=[9]-=-; Mettu and Plaxton [12] gave a simple O(n2 )-time constant approximation ratio algorithm. Thorup [14] considered the facility location problem in metric spaces defined by a graph. If the underlying g... |

266 | Approximation algorithms for facility location problems - Shmoys, Tardos, et al. - 1997 |

207 | Improved combinatorial algorithms for facility location and k-median problems
- Charikar, Guha
- 1999
(Show Context)
Citation Context ...mbinatorial optimization. The problem is known to be NP-hard and the first constant factor approximation algorithm was given by Shmoys et al. [13]. Several other approximation algorithms are given in =-=[1, 3, 8]-=-. The best approximation ratio of 1.52, is due to Madhian, Ye, and Zhang [10], while the best lower bound of 1.463 for the approximation ratio is due to Guha and Khuller [5]. The first constant factor... |

188 | Greedy strikes back: Improved facility location algorithms
- Guha, Khuller
- 1998
(Show Context)
Citation Context ...orithms are given in [1, 3, 8]. The best approximation ratio of 1.52, is due to Madhian, Ye, and Zhang [10], while the best lower bound of 1.463 for the approximation ratio is due to Guha and Khuller =-=[5]-=-. The first constant factor approximation algorithm with almost linear running time (that is, the running time of O(n2 log n)) was given by Jain and Vazirani [9]; Mettu and Plaxton [12] gave a simple ... |

121 | Improved approximation algorithms for the uncapacitated facility location problem
- Chudak, Shmoys
(Show Context)
Citation Context ...mbinatorial optimization. The problem is known to be NP-hard and the first constant factor approximation algorithm was given by Shmoys et al. [13]. Several other approximation algorithms are given in =-=[1, 3, 8]-=-. The best approximation ratio of 1.52, is due to Madhian, Ye, and Zhang [10], while the best lower bound of 1.463 for the approximation ratio is due to Guha and Khuller [5]. The first constant factor... |

119 | A new greedy approach for facility location problems
- Jain, Mahdian, et al.
- 2002
(Show Context)
Citation Context ...mbinatorial optimization. The problem is known to be NP-hard and the first constant factor approximation algorithm was given by Shmoys et al. [13]. Several other approximation algorithms are given in =-=[1, 3, 8]-=-. The best approximation ratio of 1.52, is due to Madhian, Ye, and Zhang [10], while the best lower bound of 1.463 for the approximation ratio is due to Guha and Khuller [5]. The first constant factor... |

111 | Improved approximation algorithms for metric facility location
- Mahdian, Ye, et al.
- 2002
(Show Context)
Citation Context ...t factor approximation algorithm was given by Shmoys et al. [13]. Several other approximation algorithms are given in [1, 3, 8]. The best approximation ratio of 1.52, is due to Madhian, Ye, and Zhang =-=[10]-=-, while the best lower bound of 1.463 for the approximation ratio is due to Guha and Khuller [5]. The first constant factor approximation algorithm with almost linear running time (that is, the runnin... |

79 | Sublinear time algorithms for metric space problems
- Indyk
- 1999
(Show Context)
Citation Context ...solution; results like our sublinear-time algorithm for a O(1)-factor approximation of the cost of the optimum solution for the metric uniform Minimum Facility Location problem are rare (see however, =-=[4, 6, 7]-=-). 1.2 Our Techniques Our analysis of a sublinear-time algorithm consists of two principal steps: we first prove the existence of an appropriated estimator for the cost of the Minimum Facility Locatio... |

75 | The online median problem
- Mettu, Plaxton
(Show Context)
Citation Context ...the cost of the Minimum Facility Location problem and then we show how such an estimator can be approximated in time O(n log 2 n). Our estimator is obtained by extending the primal-dual approach from =-=[12]-=-: for each point we define an approximation of the contribution of that point to the total cost, and then we prove that the sum of the contributions for all the points approximates the cost of the Min... |

44 | Approximating the minimum spanning tree weight in sublinear time
- Chazelle, Rubinfeld, et al.
- 2005
(Show Context)
Citation Context ...pling scheme to efficiently approximate the sum of the estimators. A similar approach has been used in recent sublinear-time algorithms for estimating the cost of the minimum spanning tree problem in =-=[2]-=- and [4]. 1.3 Definition of the Problem The formal definition of the general form of the (Metric) Minimum Facility Location problem is as follows: We are given a metric (P, D), and a subset F⊆P of fac... |

39 | A sublinear time approximation scheme for clustering in metric spaces
- Indyk
- 1999
(Show Context)
Citation Context ...solution; results like our sublinear-time algorithm for a O(1)-factor approximation of the cost of the optimum solution for the metric uniform Minimum Facility Location problem are rare (see however, =-=[4, 6, 7]-=-). 1.2 Our Techniques Our analysis of a sublinear-time algorithm consists of two principal steps: we first prove the existence of an appropriated estimator for the cost of the Minimum Facility Locatio... |

18 | Estimating the weight of metric minimum spanning trees in sublinear-time
- Czumaj, Sohler
- 2009
(Show Context)
Citation Context ...solution; results like our sublinear-time algorithm for a O(1)-factor approximation of the cost of the optimum solution for the metric uniform Minimum Facility Location problem are rare (see however, =-=[4, 6, 7]-=-). 1.2 Our Techniques Our analysis of a sublinear-time algorithm consists of two principal steps: we first prove the existence of an appropriated estimator for the cost of the Minimum Facility Locatio... |

16 |
Quick k-median, k-center, and facility location for sparse graphs
- Thorup
- 2001
(Show Context)
Citation Context ... for this problem was known before. It has been known that any constant factor approximation algorithm that returns not only the cost, but also a solution itself, requires the running time of Ω(n 2 ) =-=[14]-=-. Next, we prove that if the set of facilities and the cities (points that are to be connected to the facilities) are allowed to be disjoint, then any, even randomized, approximation algorithm for the... |