Improved Approximation for the Directed Spanner Problem

We prove that the size of the sparsest directed k-spanner of a graph can be approximated in polynomial time to within a factor of $\tilde{O}(\sqrt{n})$, for all k >= 3. This improves the $\tilde{O}(n^{2\mathrm{/}3})$-approximation recently shown by Dinitz and Krauthgamer.