More on Approximation Algorithms

Abstract

In this Chapter we will extend the idea which led to approximation algorithms with performance ratio 2 in order to obtain approximation algorithms with a better performance ratio for the Steiner tree problem. Recall that the main reason for the value 2 of the performance ratio of the algorithm studied so far was Lemma 6.1. It shows that the maximum ratio between the length of a minimum spanning in the complete distance network and the length of a Steiner minimum tree is at most two. Recall also that the complete distance network is just a graph on the terminal set K in which the length of each edge is equal to the length of the shortest path between the corresponding terminals in the original network N. Now observe that a shortest path between two terminals can also be viewed as a Steiner minimum tree for these two terminals. This observation motivates the following generalization: instead of just using Steiner minimum trees for all pairs of vertices we could also try to find Steiner minimum trees for all 3 or, say, r element subsets of K. This would lead to a weighted “hyper” graph instead of the complete distance network.