Computing Euclidean Steiner trees over segments

Abstract

In the classical Euclidean Steiner minimum tree (SMT) problem, we are given a set of points in the Euclidean plane and we are supposed to find the minimum length tree that connects all these points, allowing the addition of arbitrary additional points. We investigate the variant of the problem where the input is a set of line segments. We allow these segments to have length 0, i.e., they are points and hence we generalize the classical problem. Furthermore, they are allowed to intersect such that we can model polygonal input. As in the GeoSteiner approach of Juhl et al. (Math Program Comput 10(2):487–532, 2018) for the classical case, we use a two-phase approach where we construct a superset of so-called full components of an SMT in the first phase. We prove a structural theorem for these full components, which allows us to use almost the same GeoSteiner algorithm as in the classical SMT problem. The second phase, the selection of a minimal cost subset of constructed full components, is exactly the same as in GeoSteiner approach. Finally, we report some experimental results that show that our approach is more efficient than the approximate solution that is obtained by sampling the segments.