A Note on Distance-Preserving Graph Sparsification

We consider problems of the following type: given a graph $G$, how many edges are needed in the worst case for a sparse subgraph $H$ that approximately preserves distances between a given set of node pairs $P$? Examples include pairwise spanners, distance preservers, reachability preservers, etc. There has been a trend in the area of simple constructions based on the hitting set technique, followed by somewhat more complicated constructions that improve over the bounds obtained from hitting sets by roughly a $\log$ factor. In this note, we point out that the simpler constructions based on hitting sets don't actually need an extra $\log$ factor in the first place. This simplifies and unifies a few proofs in the area, and it improves the size of the $+4$ pairwise spanner from $\widetilde{O}(np^{2/7})$ [Kavitha Th. Comp. Sys. '17] to $O(np^{2/7})$.