Approximating Steiner trees and forests with minimum number of Steiner points

Let R be a finite set of terminals in a metric space (M,d). We consider finding a minimum size set S ⊆ M of additional points such that the unit-disc graph G[R ∪ S] of R ∪ S satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem G[R ∪ 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 ⊆ R × R of demand pairs and G[R ∪ S] should contains a uv-path for every uv ∈ D. Let Δ be the maximum number of points in a unit ball such that the distance between any two of them is larger than 1. It is known that Δ = 5 in ℝ². The previous known approximation ratio for ST-MSP was ⌊(Δ+1)/2⌋ +1+ϵ in an arbitrary normed space [15], and 2.5+ϵ in the Euclidean space ℝ² [5]. Our approximation ratio for ST-MSP is 1+ln(Δ−1)+ϵ in an arbitrary normed space, which in ℝ² reduces to 1+ln4+ϵ < 2.3863+ϵ. For SF-MSP we give a simple Δ- approximation algorithm, improving the folklore ratio 2(Δ−1). Finally, we generalize and simplify the Δ-approximation of Calinescu [3] for the 2-Connectivity-MSP problem, where G[R ∪ S] should be 2-connected.