A partition-based relaxation for Steiner trees

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals \({R\subseteq V}\) , and non-negative costs c _e for all edges \({e \in E}\) . Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices \({V \backslash R}\) are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J Discrete Math 19(1):122–134, 2005); it achieves a performance guarantee of \({1+\frac{\ln 3}{2}\approx 1.55}\) . The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math Program 60:145–166, 1993) and achieves an approximation ratio of 2?2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of \({G \backslash R}\) has at most b vertices. We show that Robins’ and Zelikovsky’s algorithm has an approximation ratio better than \({1+\frac{\ln 3}{2}}\) for such instances, and we prove that the integrality gap of our LP is between \({\frac{8}{7}}\) and \({\frac{2b+1}{b+1}}\) .