Computing the shortest network under a fixed topology

We show that, in any given uniform orientation metric plane, the shortest network interconnecting a given set of points under a fixed topology can be computed by solving a linear programming problem whose size is bounded by a polynomial in the number of terminals and the number of legal orientations. When the given topology is restricted to a Steiner topology, our result implies that the Steiner minimum tree under a given Steiner topology can be computed in polynomial time in any given uniform orientation metric with λ legal orientations for any fixed integer λ ≥ 2. This settles an open problem posed by Brazil, Thomas and Weng (2000).