A Polynomial Time Algorithm for Almost Optimal Vertex Fault Tolerant Spanners

We present the first polynomial time algorithm for the f vertex fault tolerant spanner problem, which achieves almost optimal spanner size. Our algorithm for constructing f vertex fault tolerant spanner takes $O(k\cdot n\cdot m^2 \cdot W)$ time, where W is the maximum edge weight, and constructs a spanner of size $O(n^{1+1/k}f^{1-1/k}\cdot (\log n)^{1-1/k})$. Our spanner has almost optimal size and is at most a $\log n$ factor away from the upper bound on the worst-case size. Prior to this work, no other polynomial time algorithm was known for constructing f vertex fault tolerant spanner with optimal size. Our algorithm is based on first greedily constructing a hitting set for the collection of paths of weight at most $k \cdot w(u,v)$ between the endpoints u and v of an edge (u,v) and then using this set to decide whether the edge (u,v) needs to be added to the growing spanner.