Abstract
In this paper, we study the following problem. Given a graph G=(V,E) and \(\ensuremath{M\underline{\subset} V}\), construct a subgraph G \(^{\rm \star}_{M}\)=(V ⋆ ,E ⋆ ) of G spanning M, with the minimum number of edges and such that for all u,v ∈ M, the distance between u and v in G and in G \(^{\rm \star}_{M}\) is the same. This is what we call an optimal partial spanner of M in G. Such a structure is “between” a Steiner tree and a spanner and could be a particularly performant and low cost structure connecting members in a network.
We prove that the problem cannot be approximated within a constant factor. We then focus on special cases: We require that the partial spanner is a tree satisfying additional conditions. For this sub problem, we describe a polynomial algorithm to construct such a tree partial spanner.
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Laforest, C. (2003). Construction of Efficient Communication Sub-structures: Non-approximability Results and Polynomial Sub-cases. In: Kosch, H., Böszörményi, L., Hellwagner, H. (eds) Euro-Par 2003 Parallel Processing. Euro-Par 2003. Lecture Notes in Computer Science, vol 2790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45209-6_124
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DOI: https://doi.org/10.1007/978-3-540-45209-6_124
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