## An Improved Approximation Ratio for the Minimum Latency Problem (1996)

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Venue: | Mathematical Programming |

Citations: | 88 - 2 self |

### BibTeX

@INPROCEEDINGS{Goemans96animproved,

author = {Michel Goemans and Jon Kleinberg},

title = {An Improved Approximation Ratio for the Minimum Latency Problem},

booktitle = {Mathematical Programming},

year = {1996},

pages = {152--158}

}

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### Abstract

Given a tour visiting n points in a metric space, the latency of one of these points p is the distance traveled in the tour before reaching p. The minimum latency problem asks for a tour passing through n given points for which the total latency of the n points is minimum; in effect, we are seeking the tour with minimum average "arrival time." This problem has been studied in the operations research literature, where it has also been termed the "delivery-man problem" and the "traveling repairman problem." The approximability of the minimum latency problem was first considered by Sahni and Gonzalez in 1976; however, unlike the classical traveling salesman problem, it is not easy to give any constant-factor approximation algorithm for the minimum latency problem. Recently, Blum, Chalasani, Coppersmith, Pulleyblank, Raghavan, and Sudan gave the first such algorithm, obtaining an approximation ratio of 144. In this work, we present an algorithm which improves this ratio to 21:55. The dev...

### Citations

355 | A general approximation technique for constrained forest problems
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ... one can obtain an (ff; fi)-T SP-approximator. Blum et al. show how to obtain a (3; 6)-T SP -approximator from a 2-approximation algorithm for the "prize-collecting TSP" due to Goemans and W=-=illiamson [11]-=-. In Section 3, when we treat the case of a general metric space, we show how a stronger analysis of the the prize-collecting TSP algorithm of [11] allows one to obtain a (2; 4)-T SP - approximator. I... |

283 |
P-Complete Approximation Problems
- Sahni, Gonzalez
- 1976
(Show Context)
Citation Context .... For weighted trees, on which the TSP is trivial, no polynomial-time algorithm is known for the MLP . Finally, although the approximability of the MLP was already considered by Sahni and Gonzalez in =-=[13]-=-, there is no simple heuristic known which gives a constant-factor approximation; the first approximation algorithm for the MLP was only recently given by Blum, Chalasani, Coppersmith, Pulleyblank, Ra... |

122 | Searching in the plane
- Baezayates, Culberson, et al.
- 1993
(Show Context)
Citation Context ...d time to find the goal. Indeed, the optimum solutions to the MLP in a number of special cases have structure similar to the on-line search patterns constructed by Baeza-Yates, Culberson, and Rawlins =-=[3]-=-. Another source of interest in this problem comes from the traveling salesman problem itself. Despite the significant differences between the TSP and MLP , the constituent parts of our approximation ... |

82 | The minimum latency problem
- Blum, Chalasani, et al.
- 1994
(Show Context)
Citation Context ...e Fellowship. Present address: Department of Computer Science, Cornell University, Ithaca NY 14853. Email: kleinber@cs.cornell.edu. On leave at IBM Almaden Research Center, San Jose CA 95120. in e.g. =-=[5, 12, 15]-=-, various local changes in the input points can lead to highly non-local changes in the optimum solution. For example, even when the input points lie on a line, the optimum MLP tour may cross itself m... |

77 |
A 3-Approximation for the Minimum Tree Spanning k Vertices
- Garg
- 1996
(Show Context)
Citation Context .... Since the publication of the conference version of this paper, two constant--factor approximations for the k-TSP have been obtained: one due to Blum, Ravi, and Vempala [7], and a second due to Garg =-=[9]-=-. Garg's algorithm provides a 3-approximation to the k-TSP , and hence constitutes a (1; 3)-T SP -approximator. Plugging these values of ff and fi into the general bound of this paper gives an approxi... |

65 | A note on the prize collecting traveling salesman problem
- Bienstock, Goemans, et al.
- 1993
(Show Context)
Citation Context ...ff and fi satisfying 2ff + fis4. However, we don't know how to exploit this fact. We can give an even better TSP -approximator by using linear programming techniques in the spirit of Bienstock et al. =-=[4]-=- for the prize-collecting TSP. Theorem 6 There exists a (2; 3)-T SP -approximator. Proof. We view V as a complete graph on n vertices, with the weight c e of the edge e = (v i ; v j ) denoting the dis... |

52 | New Approximation Guarantees for Minimum-Weight k-Tress and Prize-Collecting Salesmen - Awebuch, Azar, et al. |

48 |
Analyzing the Held-Karp TSP bound: A monotonicity property wtih application
- Shmoys, Williamson
- 1990
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Citation Context ... ! ? : 2 v i 2 W 0 v i 62 W: We now have a standard linear programming relaxation of the traveling salesman problem, written for the k vertices in W . Results of Wolsey [16] and Shmoys and Williamson =-=[14]-=- show that if we apply Christofides's heuristic to produce a tour on the vertices in W , the length of this tour will be at most 3 2 times the optimum value z k t of this linear program. Thus, we obta... |

44 | Survivable networks, linear programming relaxations and the parsimonious property
- Goemans, Bertsimas
- 1993
(Show Context)
Citation Context ...tiply our solution to (LP k ) by 1 t , obtaining x-values which satisfy x(ffi(S))s2 (v 1 2 S; W \Gamma S 6= ;): The value of the objective function is now z k t . By a result of Goemans and Bertsimas =-=[10]-=- (see also [4]), the value of the objective function is unchanged if we include the constraints x(ffi(fv i g)) = 8 ? ! ? : 2 v i 2 W 0 v i 62 W: We now have a standard linear programming relaxation of... |

42 |
Heuristic analysis, linear programming and branch and bound
- Wolsey
- 1980
(Show Context)
Citation Context ...onstraints x(ffi(fv i g)) = 8 ? ! ? : 2 v i 2 W 0 v i 62 W: We now have a standard linear programming relaxation of the traveling salesman problem, written for the k vertices in W . Results of Wolsey =-=[16]-=- and Shmoys and Williamson [14] show that if we apply Christofides's heuristic to produce a tour on the vertices in W , the length of this tour will be at most 3 2 times the optimum value z k t of thi... |

32 | A constant-factor approximation for the k-MST problem in the plane - Blum, Chalasani, et al. - 1995 |

30 | Special cases of traveling salesman and repairman problems with time windows
- Tsitsiklis
- 1992
(Show Context)
Citation Context ...e Fellowship. Present address: Department of Computer Science, Cornell University, Ithaca NY 14853. Email: kleinber@cs.cornell.edu. On leave at IBM Almaden Research Center, San Jose CA 95120. in e.g. =-=[5, 12, 15]-=-, various local changes in the input points can lead to highly non-local changes in the optimum solution. For example, even when the input points lie on a line, the optimum MLP tour may cross itself m... |

21 | The complexity of the traveling repairman problem, Informatique Théorique et Applications 20 - Afrati, Cosmadakis, et al. - 1986 |

17 |
The delivery man problem and cumulative matroids
- Fischetti, Laporte, et al.
- 1993
(Show Context)
Citation Context ...rom [5]. We are interested in approximation algorithms for this problem for a number of reasons. First of all, the MLP is a reasonably well-studied problem in the operations research literature (e.g. =-=[1, 8, 12, 13, 15]), where it is also -=-known as the "delivery-man problem" and the "traveling repairman problem." The problem is also of interest from the point of view of on-line search problems; for example, as noted ... |

16 |
The delivery man problem on a tree network
- Minieka
- 1989
(Show Context)
Citation Context ...e Fellowship. Present address: Department of Computer Science, Cornell University, Ithaca NY 14853. Email: kleinber@cs.cornell.edu. On leave at IBM Almaden Research Center, San Jose CA 95120. in e.g. =-=[5, 12, 15]-=-, various local changes in the input points can lead to highly non-local changes in the optimum solution. For example, even when the input points lie on a line, the optimum MLP tour may cross itself m... |

1 |
The complexity of the traveling repairman problem," Informatique Th'eorique et Applications
- Afrati, Cosmadakis, et al.
(Show Context)
Citation Context ...rom [5]. We are interested in approximation algorithms for this problem for a number of reasons. First of all, the MLP is a reasonably well-studied problem in the operations research literature (e.g. =-=[1, 8, 12, 13, 15]), where it is also -=-known as the "delivery-man problem" and the "traveling repairman problem." The problem is also of interest from the point of view of on-line search problems; for example, as noted ... |