Steiner trees with a terminal order

The Steiner tree problem with a terminal order (STO) consists of finding a minimum cost Steiner tree where terminals must respect a permutation. This permutation of terminal vertices gives their ranking on a real line and the minimum cost Steiner tree must be drawable in a planar way on one side of this line. A requirement like this can be found in VLSI-design problems. We propose a polynomial time algorithm to the k-Steiner tree problem with a terminal order (kSTO) and a polynomial time approximation scheme to the STO.