A greedy approximation algorithm based on spider decompositions" was developed by Klein and Ravi for node weighted Steiner trees. 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 ln k, assuming that N P6 DTIME n Olog log n , by a reduction from set cover 9, 4 . We show that for the unweighted case we can obtain an approximation factor of ln k. F or the weighted case we develop a new decomposition theorem, and generalize the notion of spiders" to branch-spiders", that are used to design a new algorithm with a worst case approximationfactor of 1:5 l n k. This algorithm, although polynomial, is not very practical due to its high running time; since we need to repeatedly nd many minimum weight matchings in each iteration. We are able to generalize the method to yield an approximation factor approaching 1:35 lnk. W e also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1:6103 lnk. The techniques developed for the second algorithm imply a method of approximating node weighted network design problems de ned by 0-1 proper functions. These new ideas also lead to improved approximation guarantees for the problem of nding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 lnn 7 . By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor approaching 1:35 ln n.
By combining a weighted set cover approximation algorithm and a nodeweighted Steiner tree approximation algorithm, they achieved the approximation ratio 3 ln n. In REF , they further improved the approximation ratio to 1:35 ln n, which is the best ratio known.
Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets
FSTTCS
Mathematics Computer Science