Tree Spanners for Subgraphs and Related Tree Covering Problems

Abstract

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.

The complexity status of deciding the existence of a Steiner tree k- spanner is easy for some k: it is $\mathcal{G}$-hard for k ≥ 4, and it is in $\mathcal{P}$ 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 $\mathcal{N}\mathcal{P}$-hardness. By showing the $\mathcal{N}\mathcal{P}$ -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.