Directed Spanners via Flow-Based Linear Programs

We examine directed spanners through flow-based linear programming relaxations. We design an (O) over tilde (n(2/3))-approximation algorithm for the directed k-spanner problem that works for all k >= 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) over tilde (n(1-1/k)) approximation [BGJ(+)09] when k >= 4. For the special case of k = 3 we design a different algorithm achieving an (O) over tilde(root n)-approximation, improving the previous (O) over tilde (n(2/3)) [EP05, BGJ(+)09] (independently of our work, an (O) over tilde (n(1-1/[k/2])) was recently devised [BRR10]). 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) over tilde (n(1/3-epsilon)) for every constant epsilon > 0. A virtue of all our algorithms is that they are relatively simple. Technically, we introduce a new yet natural flowbased 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.