Approximating Steiner Trees and Forests with Minimum Number of Steiner Points

摘要

Let R be a finite set of terminals in a metric space (M, d). We consider finding a minimum size set S {is contained in} 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 {is contained in} 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 R~2. The previous known approximation ratio for ST-MSP was [(Δ + 1)/2] + 1 + ε in an arbitrary normed space [15], and 2.5 + ∈ in the Euclidean space R~2 [5]. Our approximation ratio for ST-MSP is 1 + ln(Δ- 1) + ε in an arbitrary normed space, which in R~2 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.