## The Online Median Problem (2000)

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Venue: | In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 76 - 2 self |

### BibTeX

@INPROCEEDINGS{Mettu00theonline,

author = {Ramgopal R. Mettu and C. Greg Plaxton},

title = {The Online Median Problem},

booktitle = {In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science},

year = {2000},

pages = {339--348}

}

### Years of Citing Articles

### OpenURL

### Abstract

We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time; a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a linear-time constant-competitive algorithm for the online median problem. In addition, we present a related, though substantially simpler, linear-time constant-factor approximation algorithm for the (metric uncapacitated) facility location problem. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time.

### Citations

677 |
Online Computation and Competitive Analysis
- Borodin, El-Yaniv
- 1998
(Show Context)
Citation Context ... approach to the online median problem is to iteratively choose the point that minimizes the objective function. Greedy strategies of this kind are commonly applied in the design of online algorithms =-=[1, 9]-=-. It turns out, however, that for the online median problem, the simple strategy suggested above has an unbounded competitive ratio. We show that a modification of this strategy that we call hierarchi... |

326 | Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation
- Jain, Vazirani
(Show Context)
Citation Context ...n for thek-median problem was recently given by 1Charikar et al. [3] and is also LP-based. That work follows a sequence of bicriteria results utilizing LPbased techniques [14, 15]. Jain and Vazirani =-=[10]-=- give the first nearly linear-time combinatorial algorithms for the facility location andk-median problems, achieving approximation ratios of3and6, respectively. While the latter algorithms are combin... |

270 | Approximation algorithms for facility location problems
- Shmoys, Tardos, et al.
- 1997
(Show Context)
Citation Context ...facility location andk-median problems; here we focus on the work that is most relevant to our results. The first constant-factor approximation algorithm for facility location is due to Shmoys et al. =-=[17]-=- and is based on rounding the (fractional) solution to a linear program. Chudak [4] gives an LP-based(1+2=e)-approximation algorithm for facility location. This was the best constant factor known unti... |

221 | A Constant-Factor Approximation Algorithm for the k-Median Problem
- Charikar, Guha, et al.
(Show Context)
Citation Context ... by NSF Grant CCRâ€“9821053. Email:framgopal, plaxtong@cs.utexas.edu.1 Introduction Recently the first constant-factor approximation algorithm was discovered for thek-median problem by Charikar et al. =-=[3]-=-; in this paper, we ask whether a constant competitive ratio can be achieved for a natural online extension of thek-median problem. LetUbe a nonempty set ofnpoints and letdbe a metric distance functio... |

209 | Improved combinatorial algorithms for facility location and k-median problems
- Charikar, Guha
- 1999
(Show Context)
Citation Context ...al) solution to a linear program. Chudak [4] gives an LP-based(1+2=e)-approximation algorithm for facility location. This was the best constant factor known until the recent work of Charikar and Guha =-=[2]-=-, which establishes a slightly lower approximation ratio of1:728. The first constant-factor approximation for thek-median problem was recently given by 1Charikar et al. [3] and is also LP-based. That... |

190 | Greedy strikes back: Improved facility location algorithms
- Guha, Khuller
- 1999
(Show Context)
Citation Context ...heuristic proposed Kuehn and Hamburger [13] yields both a constant-factor approximation for the facility location problem and a bicriteria approximation for thek-median problem [11]. Guha and Khuller =-=[7]-=- showed that greedy improvement can be used as a postprocessing step to improve the approximation guarantee of certain facility location algorithms. Guha and Khuller also provide the best lower bound ... |

149 | Analysis of a local search heuristic for facility location problems - Korupolo, Plaxton |

147 |
Discrete Location Theory
- Mirchandani, Francis
- 1990
(Show Context)
Citation Context ...rithms can be easily modified to obtain a constant-competitive online median algorithm. 1.3 Contributions Algorithms for problems in discrete location theory arise in many practical applications; see =-=[5, 16]-=-, for example, for numerous pointers to the literature. Given that many of these problems are NP-hard, it is desirable to develop fast approximation algorithms. As mentioned above, it is not uncommon ... |

123 | Improved Approximation Algorithms for the Uncapacitated Facility Location Problem
- Chudak, Shmoys
(Show Context)
Citation Context ...ant to our results. The first constant-factor approximation algorithm for facility location is due to Shmoys et al. [17] and is based on rounding the (fractional) solution to a linear program. Chudak =-=[4]-=- gives an LP-based(1+2=e)-approximation algorithm for facility location. This was the best constant factor known until the recent work of Charikar and Guha [2], which establishes a slightly lower appr... |

123 | The primal-dual method for approximation algorithms and its application to network design problems
- Goemans, Williamson
- 1997
(Show Context)
Citation Context ...blems, achieving approximation ratios of3and6, respectively. While the latter algorithms are combinatorial, the primal-dual approach used in their analysis is based on linear programming theory. (See =-=[6]-=- for an excellent introduction to the primal-dual method.) Strategies based on local search and greedy techniques for facility location and thek-median problem have been previously studied. The work o... |

97 |
A Heuristic Program for Locating Warehouses
- Kuehn, Hamburger
- 1963
(Show Context)
Citation Context ... greedy techniques for facility location and thek-median problem have been previously studied. The work of Korupolu et al. [11] shows that a simple local search heuristic proposed Kuehn and Hamburger =-=[13]-=- yields both a constant-factor approximation for the facility location problem and a bicriteria approximation for thek-median problem [11]. Guha and Khuller [7] showed that greedy improvement can be u... |

79 | Sublinear time algorithms for metric space problems
- Indyk
- 1999
(Show Context)
Citation Context ...eforeO(n2+`n), which is linear in the size of the input (in bits). 104 Concluding Remarks We plan to investigate whether the ideas presented above can be applied to other problems. The work of Indyk =-=[8]-=- gives a technique to achieve sublinear time bounds for various location problems through random sampling of the distance function; we would like to see if application of these techniques to our algor... |

71 | Approximation algorithms for geometric median problems
- LIN, VITTER
- 1992
(Show Context)
Citation Context ...constant-factor approximation for thek-median problem was recently given by 1Charikar et al. [3] and is also LP-based. That work follows a sequence of bicriteria results utilizing LPbased techniques =-=[14, 15]-=-. Jain and Vazirani [10] give the first nearly linear-time combinatorial algorithms for the facility location andk-median problems, achieving approximation ratios of3and6, respectively. While the latt... |

59 | Online Computation and Competitive Analysis. Cambridge University Press, The Pitt Building, Trumpington - Borodin, El-Yaniv - 2002 |

54 | Placement Algorithms for Hierarchical Cooperative Caching
- Korupolu, Plaxton, et al.
- 1999
(Show Context)
Citation Context ... for various location problems through random sampling of the distance function; we would like to see if application of these techniques to our algorithms yield sublinear time bounds. Korupolu et al. =-=[12]-=- give an algorithm and an efficient distributed implementation for hierarchical cooperative caching in which the distance function is an ultrametric. We would like to see if the hierarchical greedy st... |

32 |
approximations with minimum packing constraint violation (extended abstract
- Lin, Vitter
- 1992
(Show Context)
Citation Context ...constant-factor approximation for thek-median problem was recently given by 1Charikar et al. [3] and is also LP-based. That work follows a sequence of bicriteria results utilizing LPbased techniques =-=[14, 15]-=-. Jain and Vazirani [10] give the first nearly linear-time combinatorial algorithms for the facility location andk-median problems, achieving approximation ratios of3and6, respectively. While the latt... |

25 | On online computation
- Irani, Karlin
- 1997
(Show Context)
Citation Context ... approach to the online median problem is to iteratively choose the point that minimizes the objective function. Greedy strategies of this kind are commonly applied in the design of online algorithms =-=[1, 9]-=-. It turns out, however, that for the online median problem, the simple strategy suggested above has an unbounded competitive ratio. We show that a modification of this strategy that we call hierarchi... |

14 | facility location, and the Chernoff-Wald bound - K-medians - 2000 |

12 | A heuristic program for locating warehouses", Management Science 9 - Kuehn, Hamburger - 1963 |

6 |
Location and Layout Planning: An International Bibliography
- Domschke, Drexl
- 1985
(Show Context)
Citation Context ...rithms can be easily modified to obtain a constant-competitive online median algorithm. 1.3 Contributions Algorithms for problems in discrete location theory arise in many practical applications; see =-=[5, 16]-=-, for example, for numerous pointers to the literature. Given that many of these problems are NP-hard, it is desirable to develop fast approximation algorithms. As mentioned above, it is not uncommon ... |