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A PartitionBased Relaxation For Steiner Trees
, 2009
"... The Steiner tree problem is a classical NPhard 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 ⊆ V, and nonnegative costs ce for all edges e ∈ E. Any tree that contains all terminals ..."
Abstract

Cited by 6 (2 self)
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The Steiner tree problem is a classical NPhard 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 ⊆ V, and nonnegative costs ce for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimumcost Steiner tree. The vertices V\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, 2005); it achieves a performance guarantee of 1 + ln3 2 ≈ 1.55. The best known linear programming (LP)based algorithm, on the other hand, is due to Goemans and Bertsimas (Math. Programming, 1993) and achieves an approximation ratio of 2 − 2/R. In this paper we establish a link between greedy and LPbased approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primaldual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the wellknown bidirected cut relaxation. An instance is bquasibipartite if each connected component of G\R has at most b vertices. We show that Robins ’ and Zelikovsky’s algorithm has an approximation ratio better than 1 + ln3 2 for such instances, and we prove that the integrality gap of our LP is between 8 7