Applications of the Matroid Parity Problem to Approximating Steiner Trees

The Steiner tree problem in graphs requires to nd a minimum size connected subgraph containing a given subset of nodes (given points). In this paper we consider this problem in three classes of graphs: where the maximum path distance is 2, where given points form a dominating set and where the given points form a vertex cover. As all these problems are MAX-SNP hard, the issue is what approximation can be obtained in polynomial time. In the rst case we obtain an approximation ratio (of the achieved size over the minimal one) 5 4 + 3 80 = 1:2875, in the second case we achieve 4/3, and in the last case we achieve 8=7 ? 1=160.