## Online Network Optimization Problems (1998)

Venue: | In Developments from a |

Citations: | 9 - 1 self |

### BibTeX

@INPROCEEDINGS{Kalyanasundaram98onlinenetwork,

author = {Bala Kalyanasundaram and Kirk Pruhs},

title = {Online Network Optimization Problems},

booktitle = {In Developments from a},

year = {1998},

pages = {268--280},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

. We survey results on online versions of the standard network optimization problems, including the minimum spanning tree problem, the minimum Steiner tree problem, the weighted and unweighted matching problems, and the traveling salesman problem. The goal in these problems is to maintain, with minimal changes, a low cost subgraph of some type in a dynamically changing network. 1 Introduction In the early 1920's Otakar Bor uvka was asked by the Electric Power Company of Western Moravia (EPCWM) to assist in EPCWM's electrification of southern Moravia by solving from a mathematical standpoint the question of how to construct the most economical electric power network [9]. In 1926 Bor uvka initiated the study of network optimization problems, by publishing an efficient algorithm for constructing a minimum spanning tree of a fixed network [9]. Certainly since the 1920's the underlying collection of sites that require electrification in southern Moravia has changed frequently as new sites ...

### Citations

143 |
Dynamic Steiner tree problem
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(Show Context)
Citation Context ...t least r(u j ; v j ) disjoint paths between u j and v j for all jsi. Note that in general the subgraph H need not be a single connected component. With the exception of the paper of Imase and Waxman =-=[12]-=-, all the papers assume the monotonic model. We use n to denote the number of nodes in the underlying network N , s to denote the number of requests, and k to denote the number of distinct vertices th... |

134 |
An optimal algorithm for on-line bipartite matching
- Karp, Vazirani, et al.
- 1990
(Show Context)
Citation Context ...matching. Notice that the competitive ratio is at most 1, and to be consistent with other papers we follow this notation instead of considering the inverse of this ratio. Karp, Vazirani, and Vazirani =-=[19]-=- consider the standard matching problem, with each server capacity c j is equal to 1, in the monotonic model. Since every maximal matching is at least half of the size of a maximum matching, every det... |

121 |
An Analysis of Several Heuristics for the Traveling Salesman Problem
- Rosenkrantz, Sterns, et al.
- 1977
(Show Context)
Citation Context ...gorithm will be universal since it uses nearest neighbor search to search configurations that have the same optimal cost. Nearest neighbor is known to be only \Theta(log n) competitive for online TSP =-=[21]-=-. Although, a proof that the generalized work function algorithm is within O(logn) of being universal would be a good first step toward finding a universal algorithm. Open Question: What is the optima... |

100 | Competitive algorithms for distributed data management
- Bartal, Fiat, et al.
- 1992
(Show Context)
Citation Context ...es optimum. Open Question: Is it possible to reduce the number of edge deletions to constant per change to G? Finally we mention some results that are closely related to GSP . Bartal, Fiat and Rabani =-=[5]-=- used competitive analysis to measure the efficiency of algorithms for the file allocation problem which is closely related to GSP . Awerbuch, Azar and Bartal [4] generalized GSP to the Network Connec... |

86 |
P.: On the history of the minimum spanning tree problem
- Graham, Hell
- 1985
(Show Context)
Citation Context ... Western Moravia (EPCWM) to assist in EPCWM's electrification of southern Moravia by solving from a mathematical standpoint the question of how to construct the most economical electric power network =-=[9]-=-. In 1926 Borsuvka initiated the study of network optimization problems, by publishing an efficient algorithm for constructing a minimum spanning tree of a fixed network [9]. Certainly since the 1920'... |

50 | On-line Steiner trees in the Euclidean plane
- Alon, Azar
- 1992
(Show Context)
Citation Context ...s reduced in 9 this construction at the cost of increasing the degrees of the nodes. Open Question: Is there an online algorithm which is close to optimal in size, diameter, and degree? Alon and Azar =-=[1]-=- considered the online Steiner tree problem where the underlying graph is the Euclidean plane. Based on a construction by Bentley and Saxe [6], they proved a lower bound of\Omega (log k= log log k) on... |

49 | Online weighted matching
- Kalyanasundaram, Pruhs
- 1993
(Show Context)
Citation Context ...erhaps allowing fi to be slightly larger than 1 might make the online algorithm's task easier. Note that the problem becomes almost trivial for fi = 2. 2.2 Weighted Matching Kalyanasundaram and Pruhs =-=[14]-=- and Khuller, Mitchell, and Vazirani [20] considered the online assignment problem, i.e. the online transportation problem where each capacity c j = 1 (so nsm). If the costs are arbitrary then [14, 20... |

40 | On-line generalized Steiner problem
- Awerbuch, Azar, et al.
- 1996
(Show Context)
Citation Context ...wed that the greedy algorithm has a competitive ratio of O( p k log k). In contrast, the competitive ratio of every deterministic online algorithm is\Omega (log k) [12, 24]. Awerbuch, Azar and Bartal =-=[4]-=- proved that the greedy algorithm is O(log 2 k) competitive. Subsequently [7] gave a O(logk) competitive online algorithm. Open Question: What is the competitive ratio of the greedy algorithm? Imase a... |

34 |
On-line algorithms for steiner tree problems (extended 117
- Berman, Coulston
- 1997
(Show Context)
Citation Context ...ntrast, the competitive ratio of every deterministic online algorithm is\Omega (log k) [12, 24]. Awerbuch, Azar and Bartal [4] proved that the greedy algorithm is O(log 2 k) competitive. Subsequently =-=[7]-=- gave a O(logk) competitive online algorithm. Open Question: What is the competitive ratio of the greedy algorithm? Imase and Waxman [12] considered the dynamic Steiner tree problem, which is a restri... |

29 | On-line algorithms for weighted bipartite matching and stable marriages
- Khuller, Mitchell, et al.
- 1994
(Show Context)
Citation Context ...than 1 might make the online algorithm's task easier. Note that the problem becomes almost trivial for fi = 2. 2.2 Weighted Matching Kalyanasundaram and Pruhs [14] and Khuller, Mitchell, and Vazirani =-=[20]-=- considered the online assignment problem, i.e. the online transportation problem where each capacity c j = 1 (so nsm). If the costs are arbitrary then [14, 20] show that no algorithm (deterministic o... |

27 | Constructing competitive tours from local information. Theoretical Computer Science 130(1):125–138
- Kalyanasundaram, Pruhs
- 1994
(Show Context)
Citation Context ... to have a competitive ratio of 7=4 on a line. Motivated by the problem of a salesman attempting to visit all the towns in some rural state for which he does not have a map, Kalyanasundaram and Pruhs =-=[15]-=- define the following version of online TSP. When the salesman visits a vertex v, it learns of each vertex w adjacent to v in N , as well as the length jvwj of the edge vw. The salesman's goal is to v... |

18 |
A Competitive Analysis of Algorithms for Searching Unknown Scenes
- Kalyanasundaram, Pruhs
- 1993
(Show Context)
Citation Context ... it travels divided by the distance between the origin and x. This corresponds to a lazy adversary that only moves one server. There is an O( p n) competitive algorithm for hide and seek in the plane =-=[13]-=- and a constant competitive 7 algorithm for the line. We believe these are also the optimal competitive ratios for online assignment in these metric spaces. One possible candidate algorithm for the li... |

18 | The online transportation problem
- Kalyanasundaram, Pruhs
(Show Context)
Citation Context ...g, [14] showed that the greedy algorithm, which matches each server to the farthest available server, is 3-competitive and that this is optimal for deterministic algorithms. Kalyanasundaram and Pruhs =-=[16]-=- consider the online transportation problem under the assumption that the capacity of each site c j is equal to some integer 6 k. The competitive ratio for greedy is still 2 m \Gamma 1, but the compet... |

11 | Serving request with on-line routing
- Ausiello, Feuerstein, et al.
- 1994
(Show Context)
Citation Context ...ithin the paradigm of search problems (see the survey on search problems) they do not fit neatly within in the models proposed in the introduction. Ausiello, Feuerstein, Leonardi, Stougie, and Talamo =-=[2, 3]-=- consider a variant of the TSP problem with release times. Let t i be the time that the ith point p i is added to G. It is required that the salesman must visit p i after time t i . One can imagine th... |

10 | Online matching with blocked input
- Kao, Tate
- 1991
(Show Context)
Citation Context ...request, Ranking selects the highest ranked unused site to service that request. [19] also show that Ranking is optimally competitive, up to lower order terms, for randomized algorithms. Kao and Tate =-=[18]-=- give some partial results for the case that points arrive in groups of t at a time. Kalyanasundaram and Pruhs [17] continue the study in the monotonic model for the case where each capacity c j is eq... |

9 |
Constructing reliable communication networks of small wight online. unpublished manuscript
- Chandra, Vishwanathan
- 1993
(Show Context)
Citation Context ...model. They showed a lower bound of\Omega (log k) on the deterministic competitive factor for the monotonic case. They showed that the greedy algorithm is O(logk)-competitive. Chandra and Viswanathan =-=[8]-=- extended the results of Imase and Waxman by considering the problem of constructing a d-connected subgraph of the revealed nodes in the monotone model. They showed that the greedy algorithm is 8d 2 l... |

7 |
An analysis of two heuristics for the euclidean travelling salesman problem
- Bentley, Saxe
- 1980
(Show Context)
Citation Context ...e to optimal in size, diameter, and degree? Alon and Azar [1] considered the online Steiner tree problem where the underlying graph is the Euclidean plane. Based on a construction by Bentley and Saxe =-=[6]-=-, they proved a lower bound of\Omega (log k= log log k) on the competitive ratio of any randomized online algorithm. The one result that does not assume the monotone model is by Imase and Waxman [12],... |

7 | Online perfect matching and mobile computing
- Grove, Kao, et al.
- 1995
(Show Context)
Citation Context ... ranked server site among the set of sites with the maximal number of unused servers among the sites that can service r j . Motivated by problems in mobile computing, Grove, Kao, Krishnan, and Vitter =-=[10]-=- consider the unweighted matching problem in essentially the fixed quality model, with the quality parameter fi = 1. That is they wish to always maintain a maximum matching. Strictly speaking, [10] as... |

5 |
Average performance of a greedy algorithm for the on-line minimum matching problem on Euclidean
- Tsai, Tang, et al.
- 1994
(Show Context)
Citation Context ...es a request with the closest unused server has a competitive ratio of 2 m \Gamma 1 [14]. Assuming that the points are uniformly scattered on the unit disk in the Euclidean plane, Tsai, Tang and Chen =-=[22]-=- showed that the expected cost for greedy is at most O( p m) times the cost of the minimum matching. Other variants of online assignment have also been considered. For the online bottleneck matching p... |

4 | Competitive Algorithms for the on-line traveling salesman, Lecture notes in computer science, Volume 955, Springer 229
- Ausiello, Feuerstein, et al.
- 1995
(Show Context)
Citation Context ...ithin the paradigm of search problems (see the survey on search problems) they do not fit neatly within in the models proposed in the introduction. Ausiello, Feuerstein, Leonardi, Stougie, and Talamo =-=[2, 3]-=- consider a variant of the TSP problem with release times. Let t i be the time that the ith point p i is added to G. It is required that the salesman must visit p i after time t i . One can imagine th... |

4 |
Linear bounds for on-line steiner problems
- Westbrook, Yan
- 1995
(Show Context)
Citation Context ...ication/transportation network of low cost. Usually, reliability means that we would like to maintain connections between given pairs of sites even if some number of sites/links are down. Krarup (see =-=[23]-=-) introduced the generalized Steiner problem GSP to study many variants of such design problems. Given an undirected graph G with positive weights on the edges, the goal in GSP is to construct a minim... |

3 |
A better lower bound for on-line bottleneck matching, manuscript
- Idury, Schaffer
- 1992
(Show Context)
Citation Context ...n considered. For the online bottleneck matching problem, where you want to minimize the maximum weight of an edge in the matching, Permutationsis still 2m \Gamma 1 competitive and Idury and Schaffer =-=[11] sho-=-wed a lower bound on the competitive ratio of any deterministic algorithm that approaches m= ln 2 �� 1:44 m for large m. For the problem of maximizing the weight of the matching, [14] showed that ... |

2 |
Lazy and greedy: On-line algorithms for Steiner problems
- Westbrook, Yan
- 1993
(Show Context)
Citation Context ...tive ratios be improved if the underlying graph is a metric space, i.e. it is complete graph and the edge weights satisfy the triangle inequality? In the case that each r(u; v) = 1, Westbrook and Yan =-=[24]-=- showed that the greedy algorithm has a competitive ratio of O( p k log k). In contrast, the competitive ratio of every deterministic online algorithm is\Omega (log k) [12, 24]. Awerbuch, Azar and Bar... |

1 |
An optimal determinstic algorithm for online b-matching. manuscript
- Kalyanasundaram, Pruhs
- 1996
(Show Context)
Citation Context ...imally competitive, up to lower order terms, for randomized algorithms. Kao and Tate [18] give some partial results for the case that points arrive in groups of t at a time. Kalyanasundaram and Pruhs =-=[17]-=- continue the study in the monotonic model for the case where each capacity c j is equal to some fixed integer k. [17] give a deterministic algorithm Balance whose competitive ratio is 1 \Gamma 1 (1+ ... |