The group Steiner tree problem consists of, given a graph $G$, a collection $\mathcal{R}$ of subsets of $V(G)$, and a positive cost $c_e$ for each edge $e$ of $G$, finding a minimum-cost tree in $G$ that contains at least one vertex from each $R \in \mathcal{R}$. We call the sets in $\mathcal{R}$ groups. The well-known Steiner tree problem is the special case of the group Steiner tree problem in which each set in $\mathcal{R}$ is unitary. In this paper, we present a general reduction test designed to remove group vertices, that is, vertices belonging to some group. Through the use of these tests we can conclude that a given group vertex can be considered a nonterminal and hence can be removed from its group. We also present some computational results on instances from SteinLib [T. Koch, A. Martin, and S. Voss, SteinLib: An updated library on Steiner tree problems in graphs, in Steiner Trees in Industry, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 285–325].
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SIAM Journal on Optimization
Networks and Optimization

de Oliveira Filho, F.M, & Ferreira, C.E. (2007). New reduction techniques for the group Steiner tree problem. SIAM Journal on Optimization, 17(4), 1176–1188.