On Approximations for Constructing 1-Line Minimum Rectilinear Steiner Trees in the Euclidean Plane $$\mathbb {R}^2$$

Abstract

In this paper, we consider the 1-line minimum rectilinear Steiner tree (1L-MRST) problem, which is defined as follows. Given n points in the Euclidean plane \(\mathbb {R}^2\), we are asked to find the location of a line l and a Steiner tree T(l), which consists of vertical and horizontal line segments plus the line l, to interconnect these n points and at least one point on the line l, the objective is to minimize total weight of T(l), i.e., \(\min \{\sum _{uv\in T(l)} w(u,v)\) | T(l) is a Steiner tree as mentioned-above\(\}\), where weight \(w(u,v)=0\) if two endpoints u, v of an edge \(uv \in T(l)\) is located on the line l and weight w(u, v) as the rectilinear distance between u and v otherwise. Given a line l as an input, we refer to this problem as the 1-line-fixed minimum rectilinear Steiner tree (1LF-MRST) problem; In addition, when Steiner points of T(l) are all located on the line l, we refer to this problem problem as the constrained minimum rectilinear Steiner tree (CMRST) problem.