The clustered selected-internal Steiner tree problem

Given a complete graph $G=(V,E)$, with nonnegative edge costs, two subsets $R \subset V$ and $R^{\prime} \subset R$, a partition $\mathcal{R}=\{R_1,R_2,\ldots,R_k\}$ of $R$, $R_i \cap R_j=\phi$, $i \neq j$ and $\mathcal{R}^{\prime}=\{R^{\prime}_1,R^{\prime}_2,\ldots,R^{\prime}_k\}$ 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.