Abstract
Let \(R\) be a finite set of terminals in a metric space \((M,d)\). We consider finding a minimum size set \(S \subseteq M\) of additional points such that the unit-disc graph \(G[R \cup S]\) of \(R \cup S\) satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem \(G[R \cup S]\) should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-MSP) problem we are given a set \(D \subseteq R \times R\) of demand pairs and \(G[R \cup S]\) should contains a \(uv\)-path for every \(uv \in D\). Let \(\varDelta \) be the maximum number of points in a unit ball such that the distance between any two of them is larger thanĀ \(1\). It is known that \(\varDelta =5\) in \(\mathbb {R}^2\). The previous known approximation ratio for ST-MSP was \(\lfloor (\varDelta +1)/2 \rfloor +1+\epsilon \) in an arbitrary normed space [15], and \(2.5+\epsilon \) in the Euclidean space \(\mathbb {R}^2\) [5]. Our approximation ratio for ST-MSP is \(1+\ln (\varDelta -1)+\epsilon \) in an arbitrary normed space, which in \(\mathbb {R}^2\) reduces to \(1+\ln 4+\epsilon < 2.3863 +\epsilon \). For SF-MSP we give a simple \(\varDelta \)-approximation algorithm, improving the folklore ratio \(2(\varDelta -1)\). Finally, we generalize and simplify the \(\varDelta \)-approximation of Calinescu [3] for the \(2\) -Connectivity-MSP problem, where \(G[R \cup S]\) should be \(2\)-connected.
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Cohen, N., Nutov, Z. (2015). Approximating Steiner Trees and Forests with Minimum Number of Steiner Points. In: Bampis, E., Svensson, O. (eds) Approximation and Online Algorithms. WAOA 2014. Lecture Notes in Computer Science(), vol 8952. Springer, Cham. https://doi.org/10.1007/978-3-319-18263-6_9
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