TY - GEN

T1 - An approximation algorithm for the Group Steiner Problem

AU - Even, Guy

AU - Kortsarz, Guy

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2002

Y1 - 2002

N2 - The input in the Group-Steiner Problem consists of an undirected connected graph with a cost function p(e) over the edges and a collection of subsets of vertices {gi} Each subset gi is called a group and the vertices in Ugi are called terminals. The goal is to find a minimum cost tree that contains at least one terminal from every group. We give the first combinatorial polylogarith-mic ratio approximation for the problem on trees. Let m denote the number of groups and S denote the number of terminals. The approximation ratio of our algorithm is O(logS · log m/log log S) = O(log2n/loglogn). This is an improvement by a φ(log log n) factor over the previously best known ap proximation ratio for the Group Steiner Problem on trees [GKR98]. Our result carries over to the Group Steiner Problem on general graphs and to the Covering Steiner Problem. Garg et al. [GKR98] presented a reduction of the Group Steiner Problem on general graphs to trees. Their reduction employs Bar-tal's [B98] approximation of graph metrics by tree metrics. Our algorithm on trees implies an approximation algorithm of ratio O(logS · logm · logn · log log n/ log log S) = O(log3 n) for the Group Steiner Problem on general graphs. The previously best known approximation ratio for this problem on general graphs, as a function of n, is O(log3n · log logn) [GKR98]. Our algorithm in conjunction with ideas of [EKS01] gives an O(logS · logm· logn· log logn/ log log S) = O(log3 n)-approximation ratio for the more general Covering Steiner Problem, improving the best known approximation ratio (as a function of n) for the Covering Steiner Problem by a Φ(loglogn) factor.

AB - The input in the Group-Steiner Problem consists of an undirected connected graph with a cost function p(e) over the edges and a collection of subsets of vertices {gi} Each subset gi is called a group and the vertices in Ugi are called terminals. The goal is to find a minimum cost tree that contains at least one terminal from every group. We give the first combinatorial polylogarith-mic ratio approximation for the problem on trees. Let m denote the number of groups and S denote the number of terminals. The approximation ratio of our algorithm is O(logS · log m/log log S) = O(log2n/loglogn). This is an improvement by a φ(log log n) factor over the previously best known ap proximation ratio for the Group Steiner Problem on trees [GKR98]. Our result carries over to the Group Steiner Problem on general graphs and to the Covering Steiner Problem. Garg et al. [GKR98] presented a reduction of the Group Steiner Problem on general graphs to trees. Their reduction employs Bar-tal's [B98] approximation of graph metrics by tree metrics. Our algorithm on trees implies an approximation algorithm of ratio O(logS · logm · logn · log log n/ log log S) = O(log3 n) for the Group Steiner Problem on general graphs. The previously best known approximation ratio for this problem on general graphs, as a function of n, is O(log3n · log logn) [GKR98]. Our algorithm in conjunction with ideas of [EKS01] gives an O(logS · logm· logn· log logn/ log log S) = O(log3 n)-approximation ratio for the more general Covering Steiner Problem, improving the best known approximation ratio (as a function of n) for the Covering Steiner Problem by a Φ(loglogn) factor.

UR - http://www.scopus.com/inward/record.url?scp=26444605118&partnerID=8YFLogxK

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AN - SCOPUS:26444605118

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 49

EP - 58

BT - Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

PB - Association for Computing Machinery

T2 - 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002

Y2 - 6 January 2002 through 8 January 2002

ER -