Tree Spanners for Subgraphs and Related Tree Covering Problems

For any fixed parameter k ≥ 1, a tree k–spanner of a graph G is a spanning tree T in G such that the distance between every pair of vertices in T is at most k times their distance in G. In this paper, we generalize on this very restrictive concept, and introduce Steiner tree k–spanners: We are given an input graph consisting of terminals and Steiner vertices, and we are now looking for a tree k–spanner that spans all terminals. We study the problems of deciding whether such a Steiner tree k–spanner exists in a given graph as well as finding a smallest Steiner tree k–spanner (if one exists at all). The complexity status of deciding the existence of a Steiner tree k–spanner is easy for some k: it is NP-hard for k ≥ 4, and it is in P 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 NP-hardness. By showing the NP-hardness also for the case k = 3, the complexity results for all k are complete. If we know in advance that a given graph contains a Steiner tree k–spanner for any arbitrary k ≥ 2, we prove that we cannot even 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.