Augmenting Graphs to Minimize the Radius

We study the problem of augmenting a metric graph by adding k edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time (5 / 3 − ϵ )-approximation algorithm, for any ϵ > 0, unless P = NP . We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth.