Constrained spanning, Steiner trees and the triangle inequality

We consider the approximation characteristics of constrained spanning and Steiner tree problems in weighted undirected graphs where the edge costs and delays obey the triangle inequality. The constraint here is in the number of hops a message takes to reach other nodes in the network from a given source. A hop, for instance, can be a message transfer from one end of a link to the other. A weighted hop refers to the amount of delay experienced by a message packet in traversing the link. The main result of this chapter shows that no approximation algorithm for a delay-constrained spanning tree satisfying the triangle inequality can guarantee a worst case approximation ratio better than Θ(log n) unless NP ⊂ DTIME(n log logn ). This result extends to the corresponding problem for Steiner trees which satisfy the triangle inequality as well.