A greedy approximation algorithm for the group Steiner problem

  • Authors:

  • Chandra Chekuri;Guy Even;Guy Kortsarz

  • Affiliations:

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

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

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Abstract

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices {g"i}"i"="1^m. Each subset g"i is called a group and the vertices in @?"ig"i 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 @e0, our algorithm gives an O((log@?"i|g"i|)^1^+^@e.logm) 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(max"i|g"i|).logm) provided by the LP based approaches. 1996, pp. 184-93, On approximating arbitrary metrics by tree metrics, Proceedings of STOC, 1998, pp. 161-168, A tight bound on approximating arbitrary metrics by tree metrics, Proceedings of STOC, 2003, pp. 448-455]. 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(max"i|g"i|).logm) provided by the LP based approaches [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].