Approximating Low-Stretch Spanners

Despite significant recent progress on approximating graph spanners (subgraphs which approximately preserve distances), there are still several large gaps in our understanding. We give new results for two of them: approximating basic k-spanner (particularly for small k), and the dependence on f when approximating f-fault tolerant spanners. We first design an O(n1/3)-approximation for 4-spanner (both basic and directed). This was the last value of k for which only an O([EQUATION])-approximation was known for basic k-spanner, and thus implies that for any k the approximation ratio is at most O(n1/3). For basic k-spanner, we also show an integrality gap for the natural flow-based LP (the main tool in almost all nontrivial spanner approximations) which nearly matches the trivial approximation of n [EQUATION]. For f-fault tolerant spanners, we show that in the small-stretch setting (k ∈ {3, 4}) it is possible to entirely remove the dependence on f from the approximation ratio, at the cost of moving to bicriteria guarantees. The previous best dependence on f was either almost-linear (in the undirected setting) or exponential (in the directed setting for stretch 4).