Routing Problems in Multicast Networks

The approximability characteristics of the constrained Steiner tree (CST) problem and some of 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 have attempted to classify problems as to whether they can be in APX or not. In chapter 2, we first show that two special cases of CST, hop-constrained spanning tree and hop-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, hop-constrained spanning tree with edge weights 1 or 2, cannot have a polynomial time approximation scheme unless P = NP. Chapter 3 presents an (exact) branch and bound algorithm as well as two heuristics, Selection Search and the Mixed Priority Search, based on search trees for CST. These heuristics are similar to Dijkstra's and Prim's algorithms in that they build a tree beginning at the source and add an edge and a node to the tree at each iteration. These run faster than any multicasting heuristic known so far, in times $O(m+n\ {\rm log}\ n)$ and O(m log n) respectively. We then analyze the performance of these heuristics with respect to parameters like cost ratio (the ratio of the cost of the most expensive edge to that of the least expensive edge) and graph density. Chapter 4 gives conclusions and suggestions for further research.