Approximation algorithm for solving the 1-line Steiner tree problem with minimum number of Steiner points

We address the 1-line Steiner tree problem with minimum number of Steiner points. Given a line l, a point set P of n terminals in R-2 and a positive constant K, we are asked to find a Steiner tree T-l to interconnect the line l and the n terminals such that the Euclidean length of each edge in T-l is no more than the given positive constant K except those connecting two points on the line l, the objective is to minimize total number of the Steiner points in T-l, i.e. min T-l {vertical bar S-out vertical bar + vertical bar S-on vertical bar}, where vertical bar S-out vertical bar and vertical bar S-on vertical bar are the number of Steiner points located outside of the line l and on this line l, respectively. We design a 4-approximation algorithm with time complexity of O(n(3)) for the 1-line Steiner tree problem with minimum number of Steiner points.