We establish significantly improved bounds on the performance of the greedy algorithm for approximating set cover. In particular, we provide the first substantial improvement of the 20 year old classical harmonic upper bound, H(m), of Johnson, Lovasz, and Chv'atal, by showing that the performance ratio of the greedy algorithm is, in fact, exactly ln m \Gamma ln ln m+ \Theta(1), where m is the size of the ground set. The difference between the upper and lower bounds turns out to be less than 1:1. This provides the first tight analysis of the greedy algorithm, as well as the first upper bound that lies below H(m) by a function going to infinity with m. We also show that the approximation guarantee for the greedy algorithm is better than the guarantee recently established by Srinivasan for the randomized rounding technique, thus improving the bounds on the integrality gap. Our improvements result from a new approach which might be generally useful for attacking other similar problems. ...

The approximability characteristics of the constrained Steiner tree (CST) problem and some if its special cases are considered here. APX is the class of problems for which it is possible to have polynomial time heuristics that guarantee a constant approximation bound. We first show that two special cases of CST, height-constrained spanning tree and height-constrained Steiner tree with unit-weight edges, cannot be in APX. This implies that CST cannot be in APX either. We then show that a more restricted special case of CST, height-constrained spanning tree with edge weights 1 or 2, cannot have a polynomial time approximation scheme unless P=NP. Key words: Approximability, Constrained Steiner tree, E-reduction, Multicasting. 1 Multicasting The optimization problems considered here arise in the context of multicast routing in communication networks. Multicasting is the transmission of data from a source to a given set of destinations. Specifically, it involves sending messages to a subse...