A greedy approximation algorithm for the group Steiner problem

  • Authors:

  • Chandra Chekuri;Guy Even;Guy Kortsarz

  • Affiliations:

  • Lucent Bell Labs, Murray Hill, New Jersey;Department of Electrical-Engineering, Tel-Aviv University, Israel;Computer Sciences Department, Rutgers University, Camden

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2006

Quantified Score

Hi-index 0.04

Visualization

Abstract

In the group Steiner problem we are given an edge-weighted graph G = (V, E, w) and m subsets of vertices {gi}i=1m. Each subset gi is called a group and the vertices in ∪igi are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε 0, our algorithm gives an O((log Σi|gi|1+εċ log m) approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of O(log |V|) in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(maxi|gi|)ċlog m) provided by the LP based approaches.