k-Steiner-minimal-trees in metric spaces Original Research Article

Let N be a finite set in a metric space. A Steiner-minimal-tree (SMT) for N is a tree interconnecting the points of N with shortest possible length. We introduce a k-SMT in a way that we allow at most k additional points (Steiner points) in the tree, where k⩾0 is a given integer. We make two assumptions on the ‘geometry’ of the space: 1. There is a natural number c, depending on the space only, such that the vertex-degree is at most c for any Steiner point in each k-SMT; and 2. For each number n between 3 and kc−k+1 there is an algorithm Sn for finding a shortest tree for each finite set with n points. Then in any metric space which fulfills both assumptions a k-SMT for a finite set of points can be found by a procedure in polynomially bounded time. The relative defect going from a (k−1)-SMT to a k-SMT tends to zero, when k runs to infinity.