Beyond Steiner's problem: a VLSI oriented generalization
WG '89 Proceedings of the fifteenth international workshop on Graph-theoretic concepts in computer science
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
A polylogarithmic approximation algorithm for the group Steiner tree problem
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Approximating the weight of shallow Steiner trees
Discrete Applied Mathematics
Approximation algorithms for directed Steiner problems
Journal of Algorithms
An approximation algorithm for the covering Steiner problem
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Approximating a Finite Metric by a Small Number of Tree Metrics
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Provably Good Routing Tree Construction with Multi-Port Terminals
Provably Good Routing Tree Construction with Multi-Port Terminals
Integrality ratio for group Steiner trees and directed steiner trees
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Polylogarithmic inapproximability
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A greedy approximation algorithm for the group Steiner problem
Discrete Applied Mathematics
Approximation algorithms for group prize-collecting and location-routing problems
Discrete Applied Mathematics
A greedy approximation algorithm for the group Steiner problem
Discrete Applied Mathematics
On trip planning queries in spatial databases
SSTD'05 Proceedings of the 9th international conference on Advances in Spatial and Temporal Databases
Hi-index | 0.00 |
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 ∪ gi 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 polylogarithmic 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(log S · log m/log log S) = O(log2n/log log n). This is an improvement by a Θ(log log n) factor over the previously best known approximation 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 Bartal's [B98] approximation of graph metrics by tree metrics. Our algorithm on trees implies an approximation algorithm of ratio O(log S · log m · log n · log log n/log log S) = O(log3n) 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 log n) [GKR98]. Our algorithm in conjunction with ideas of [EKS01] gives an O(log S · log m · log n · log log n/log log S) = O(log3n)-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 Θ(log log n) factor.