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

Given a complete graph G=(V,E)G=(V,E) with a metric cost function c:E→R+c:E→R+ and two vertex subsets R⊂VR⊂V and R′⊆RR′⊆R, a partial-terminal Steiner tree is a Steiner tree which contains all the vertices in R   such that all the vertices in R′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}{1,2}, namely, the (1,2)(1,2)-partial-terminal Steiner tree problem (PTSTP(1,2))(PTSTP(1,2)). In this paper, we consider PTSTP(1,2)PTSTP(1,2). The previous best-known approximation ratio of PTSTP(1,2)PTSTP(1,2) was at most 2514. In this paper, we propose a polynomial-time approximation algorithm that improves the approximation ratio from 2514 to 53.