In a directed graph G with non-correlated edge lengths and costs, the network design problem with bounded distances asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria (2+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most 2+\varepsilon.

In the course of proving this result, the related problem of directed shallow-light Steiner trees arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a (1+\varepsilon,O(|R|^{\varepsilon}))-ap\-proximation, where R is the set of terminals.

Finally, we show how to apply our results to obtain an (\alpha+\varepsilon,O(n^{0.5+\varepsilon}))-approximation for light-weight directed \alpha-spanners. For this, no non-trivial approximation algorithm has been known before. All running times depends on n and

\varepsilon and are polynomial in n for any fixed \varepsilon>0.