We investigate a practical variant of the well-known graph Steiner tree problem. For a complete graph G = ( V,E ) with length function l:E → R + and two vertex subsets R ⊂ V and R' ⊆ R, a partial terminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R \ R' belong to the leaves of this Steiner tree. The partial terminal Steiner tree problem is to find a partial terminal Steiner tree T whose total lengths ∑(u,v)∈Tl(u,v) is minimum. In this paper, we show that the problem is both NP-complete and MAX SNP-hard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem.
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