An improved approximation algorithm for the partial-terminal Steiner tree problem with edge cost 1 or 2

Given a complete graph G = ( V , E ) with a metric cost function c : E ? R + and two vertex subsets R ? V and R ' ? R , a partial-terminal Steiner tree is a Steiner tree which contains all the vertices in R such that all the vertices in R ' are leaves. The partial-terminal Steiner tree problem (PTSTP) is to find a partial-terminal Steiner tree with the minimum cost. The problem has been shown to be NP-hard and MAX SNP-hard, even when the edge costs are restricted in { 1 , 2 } , namely, the ( 1 , 2 ) -partial-terminal Steiner tree problem ( PTSTP ( 1 , 2 ) ) . In this paper, we consider PTSTP ( 1 , 2 ) . The previous best-known approximation ratio of PTSTP ( 1 , 2 ) was at most 25 14 . In this paper, we propose a polynomial-time approximation algorithm that improves the approximation ratio from 25 14 to 5 3 .

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