Given a complete graph G=(V,E), with nonnegative edge costs, two subsets R \subset V and R^{\prime} \subset R, a partition R=\{R_1,R_2,\ldots,R_k\} of R, R_i \cap R_j=\phi, i \neq j and R^{\prime}=\{R^{\prime}_1,R^{\prime}_2,\ldots,R^{\prime}_k\} of R^{\prime}, R^{\prime}_i \subset R_i, a clustered Steiner tree is a tree T of G that spans all vertices in R such that T can be cut into k subtrees T_i by removing k-1 edges and each subtree T_i spanning all vertices in R_i, 1 \leq i \leq k. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of G is a clustered Steiner tree for R if all vertices in R^{\prime}_i are internal vertices of T_i, 1 \leq i \leq k. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree T for R and R^{\prime} in G with minimum cost. It is not hard to see the clustered selected-internal Steiner tree problem is NP-hard. Hence, in this paper, we present the first known approximation algorithm with performance ratio  \rho+4  for the clustered selected-internal Steiner tree problem if the cost function is metric (i.e., the costs of edges satisfy the triangle inequality), where \rho is the best-known performance ratio for the Steiner tree problem whose performance ratio is \ln 4%DIFDELCMD < }%%% +\epsilon\approx 1.39 .
Given a complete graph G=(V,E), with nonnegative edge costs, two subsets R \subset V and R^{\prime} \subset R, a partition R=\{R_1,R_2,\ldots,R_k\} of R, R_i \cap R_j=\phi, i \neq j and R^{\prime}=\{R^{\prime}_1,R^{\prime}_2,\ldots,R^{\prime}_k\} of R^{\prime}, R^{\prime}_i \subset R_i, a clustered Steiner tree is a tree T of G that spans all vertices in R such that T can be cut into k subtrees T_i by removing k-1 edges and each subtree T_i spanning all vertices in R_i, 1 \leq i \leq k. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of G is a clustered Steiner tree for R if all vertices in R^{\prime}_i are internal vertices of T_i, 1 \leq i \leq k. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree T for R and R^{\prime} in G with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio ( \rho+4 ) for the clustered selected-internal Steiner tree problem , where \rho is the best-known performance ratio for the Steiner tree problem %DIFDELCMD < }%%% .
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if the cost function is metric (i.e., the costs of edges satisfy the triangle inequality),
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