The uniform orientation Steiner tree problem is NP-hard: International Journal of Computational Geometry & Applications

Given a set of n points (known as terminals) and a set of ? ≥ 2 uniformly distributed (legal) orientations in the plane, the uniform orientation Steiner tree problem asks for a minimum-length network that interconnects the terminals with the restriction that the network is composed of line segments using legal orientations only. This problem is also known as the ?-geometry Steiner tree problem. We show that for any fixed ? > 2 the uniform orientation Steiner tree problem is NP-hard. In fact we prove a strictly stronger result, namely that the problem is NP-hard even when the terminals are restricted to lying on two parallel lines.