The Depth-Restricted Rectilinear Steiner Arborescence Problem is NP-complete

In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions d(t) ∈ N for every terminal t. We are looking for a shortest binary rectilinear Steiner arborescence such that each terminal t is at depth d(t), that is, there are exactly d(t) Steiner points on the unique root-t-path is exactly d(t). We prove that even this restricted version is NP-hard. keyword: Steiner arborescence, depth restrictions, NP-completeness, VLSI design, shallow light Steiner trees