Directed Spanners via Flow-Based Linear Programs

We examine directed spanners through flow-based linear programming relaxations. We design an $\~O(n^{2/3})$-approximation algorithm for the directed $k$-spanner problem that works for all $k\geq 1$, which is the first sublinear approximation for arbitrary edge-lengths. Even in the more restricted setting of unit edge-lengths, our algorithm improves over the previous $\~O(n^{1-1/k})$ approximation of Bhattacharyya et al. when $k\ge 4$. For the special case of $k=3$ we design a different algorithm achieving an $\~O(\sqrt{n})$-approximation, improving the previous $\~O(n^{2/3})$. Both of our algorithms easily extend to the fault-tolerant setting, which has recently attracted attention but not from an approximation viewpoint. We also prove a nearly matching integrality gap of $\Omega(n^{\frac13 - \epsilon})$ for any constant $\epsilon > 0$. A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flow-based relaxation, and show how to approximately solve it even when its size is not polynomial. The main challenge is to design a rounding scheme that "coordinates" the choices of flow-paths between the many demand pairs while using few edges overall. We achieve this, roughly speaking, by randomization at the level of vertices.