2.1 Steiner Tree

We first show that we can assume without loss of generality that G is a complete graph. In particular, let G denote the metric completion of G: the cost of an edge (u, v) in G for all u, v ∈ V is given by the length of the shortest path between u and v in G. As an exercise verify that costs in G form a metric. Lemma 2.1.1 Let T be a tree spanning R in G, then we can find a subtree in G, T ′ that spans R and has cost at most the cost of T in G. Proof: We can get T ′ by taking a union of the paths represented by edges in T . Cost of edges in G is the same as the cost of the paths (shortest) of G they represent. Hence the sum of costs of all the paths making up T ′ is at most the cost of T .