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)∈T l(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.
The internal Steiner tree problem Design and analysis of algorithms a b s t r a c t Given a graph G = (V , E) with a cost function c : E → R + and a vertex subset R ⊂ V , an internal Steiner tree is a Steiner tree that contains all the vertices in R, and such that each vertex in R must be an internal vertex. The internal Steiner tree problem involves finding an internal Steiner tree T whose total costIn this paper, we first show that the internal Steiner tree problem is MAX SNP-hard. We then present a (2ρ + 1)-approximation algorithm for solving the problem on complete graphs, where ρ is an approximation ratio for the Steiner tree problem. Currently, the best-known ρ is ln 4+ϵ < 1.39. Moreover, for the case where the cost of each edge is restricted to being either 1 or 2, we present a 9 7 -approximation algorithm for the problem.
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