    Approximating the clustered selected-internal Steiner tree problem

Given a complete graph G=(V,E), with nonnegative edge costs, two subsets R ⊂ V and R^'⊂ R, a partition ℛ={R_1,R_2,…,R_k} of R, R_i ∩ R_j=ϕ, i ≠ j and ℛ^'={R^'_1,R^'_2,…,R^'_k} of R^', R^'_i ⊂ 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 ≤ i ≤ 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^'_i are internal vertices of T_i, 1 ≤ i ≤ 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^' in G with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio (ρ+4) for the clustered selected-internal Steiner tree problem, where ρ is the best-known performance ratio for the Steiner tree problem.

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