An approximation algorithm for the Group Steiner Problem

Guy Even, Guy Kortsarz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationProceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002
PublisherAssociation for Computing Machinery
Pages49-58
Number of pages10
ISBN (Electronic)089871513X
StatePublished - Jan 1 2002
Event13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002 - San Francisco, United States
Duration: Jan 6 2002Jan 8 2002

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume06-08-January-2002

Other

Other13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002
CountryUnited States
CitySan Francisco
Period1/6/021/8/02

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Even, G., & Kortsarz, G. (2002). An approximation algorithm for the Group Steiner Problem. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002 (pp. 49-58). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 06-08-January-2002). Association for Computing Machinery.