A greedy approximation algorithm for the group Steiner problem

• Published in: Discrete Applied Mathematics Vol. 154; no. 1; pp. 15 - 34
• Main Authors: Chekuri, Chandra, Even, Guy, Kortsarz, Guy
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 2006; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation algorithm; Combinatorial; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph theory; Greedy; Group Steiner problem; Mathematics; Sciences and techniques of general use; Theoretical computing; Tree; Tree; Combinatorial; Approximation algorithm; Greedy; Group Steiner problem; Edge(graph); Tree(graph); Computer theory; Linear programming; Steiner tree problem; Recursive algorithm; Polynomial time; Direct problem; Relaxation; Greedy algorithm; Combinatorics
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2005.07.010

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 ⋃ i g 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 ε > 0 , our algorithm gives an O ( ( log ∑ i | g i | ) 1 + ε · 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 ( max i | g i | ) · log m ) provided by the LP based approaches.