Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets

In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k, where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1 − o(1))ln k, assuming that \(NP \not\subseteq DTIME[n^{O(\log \log n)}]\), by a reduction from set cover.