# Approximation algorithms for priority Steiner tree problems

In the Priority Steiner Tree (PST) problem, we are given an undirected graph G=(V,E) with a source s ∈ V and terminals T ⊆ V ∖{s}, where each terminal v ∈ T requires a nonnegative priority P(v). The goal is to compute a minimum weight Steiner tree containing edges of varying rates such that the path from s to each terminal v consists of edges of rate greater than or equal to P(v). The PST problem with k priorities admits a min{2 ln |T| + 2, kρ}-approximation [Charikar et al., 2004], and is hard to approximate with ratio c loglog n for some constant c [Chuzhoy et al., 2008]. In this paper, we first strengthen the analysis provided by [Charikar et al., 2004] for the (2 ln |T| + 2)-approximation to show an approximation ratio of ⌈log_2 |T| ⌉ + 1 ≤ 1.443 ln |T| + 2, then provide a very simple, parallelizable algorithm which achieves the same approximation ratio. We then consider a more difficult node-weighted version of the PST problem, and provide a (2 ln |T|+2)-approximation using extensions of the spider decomposition by [Klein & Ravi, 1995]. This is the first result for the PST problem in node-weighted graphs. Moreover, the approximation ratios for all above algorithms are tight.

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