The complexity status of deciding the existence of a Steiner tree k- spanner is easy for some k: it is -hard for k ≥ 4, and it is in for k = 1. For the case k = 2, we develop a model in terms of an equivalent tree covering problem, and use this to show -hardness. By showing the -hardness also for the case k = 3, the complexity results for all k are complete.

We also consider the problem of finding a smallest Steiner tree k-spanner (if one exists at all). For any arbitrary k ≥ 2, we prove that we cannot hope to find efficiently a Steiner tree k-spanner that is closer to the smallest one than within a logarithmic factor. We conclude by discussing some problems related to the model for the case k = 2.