The relationship among some variables of Steiner tree problem

Abstract

Given a weighted complete graph G = (V, E; w) and a subset R V, the Steiner tree problem (STP) is to find a minimum sub-tree of G interconnecting R. In this paper, we consider the relationship among the following 4 variables of STP: terminal Steiner tree problem (TSTP), partial-terminal Steiner tree problem (PTSTP), internal Steiner tree problem (ISTP) and selected-internal Steiner tree problem (SISTP). Firstly, we show that if SISTP has an approximation problem with approximation ratio , then ISTP also has an approximation algorithm with the same ratio. Then, we prove that the same result holds between PTSTP and TSTP.