A note on distance-preserving graph sparsification

Abstract

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 O˜(np2/7) (Kavitha (2017) [13]) to O(np2/7).