Bypassing the surface embedding: approximation schemes for network design in minor-free graphs

Since the mid 90s, the study of the complexity of classic network design problems such as the traveling salesman problem (TSP), the Steiner tree problem (ST), or the k-MST problem on metric spaces such as low-dimensional Euclidean spaces, doubling metrics, planar or minor-free graphs, has led to major improvements of our understanding of the structure of both these important metric spaces, and the underlying problems.