We examine directed spanners through flow-based linear programming relaxations. We design anÕ(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 previ- ). 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 ofΩ(n 1/3− ) for every constant > 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.
Dinitz et al. REF provide a flow-based linear programming (LP) relaxation to approximate the directed spanner problem.
Directed spanners via flow-based linear programs
STOC '11
Mathematics Computer Science