  We consider the problem of adding a fixed number of new edges to an
undirected graph in order to minimize the diameter of the augmented graph, and
under the constraint that the number of edges added for each vertex is bounded
by an integer. The problem is motivated by network-design applications, where
we want to minimize the worst case communication in the network without
excessively increasing the degree of any single vertex, so as to avoid
additional overload. We present three algorithms for this task, each with their
own merits. The special case of a matching augmentation, when every vertex can
be incident to at most one new edge, is of particular interest, for which we
show an inapproximability result, and provide bounds on the smallest achievable
diameter when these edges are added to a path. Finally, we empirically evaluate
and compare our algorithms on several real-life networks of varying types.
