Directed spanners via flow-based linear programs

• Authors: Michael Dinitz;Robert Krauthgamer
• Venue: Proceedings of the forty-third annual ACM symposium on Theory of computing
• Year: 2011

We examine directed spanners through flow-based linear programming relaxations. We design an ~O(n2/3)-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(n1-1/k) approximation [BGJRW09] when k ≥ 4. For the special case of k=3 we design a different algorithm achieving an ~O(√n)-approximation, improving the previous ~O(n2/3) [EP05,BGJRW09] (independently of our work, an ~O(n1-1/⌈ 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 ~Ω(n1/3 - ε) for every constant ε 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.