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.
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Handke, D., Kortsarz, G. (2000). Tree Spanners for Subgraphs and Related Tree Covering Problems. In: Brandes, U., Wagner, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2000. Lecture Notes in Computer Science, vol 1928. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40064-8_20
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DOI: https://doi.org/10.1007/3-540-40064-8_20
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