# Reachability preservers: New extremal bounds and approximation algorithms

## Abstract

In this paper we prove new results about the extremal structure of paths in directed graphs. Say we are given a directed graph G = (V;E) on n nodes, a set of sources S V of size |S| = n1/3, and a subset P S V of pairs (s; t) where s 2 S, of size O(n2=3), such that for all pairs (s; t) 2 P, there is a path from s to t. Our goal is to remove as many edges from G as possible while maintaining the reachability of all pairs in P. How many edges will we have to keep? Can you always go down to n1+o(1) edges? Or maybe for some nasty graphs G you cannot even go below the simple bound of O(n4=3) edges? Embarrassingly, in a world where graph reachability is ubiquitous in countless scientific fields, the current bounds on the answer to this question are far from tight. In this paper, we make polynomial progress in both the upper and lower bounds for these Reachability Preservers over bounds that were implicit in the literature. We show that in the above scenario, O(n) edges will always be sufficient, and in general one is even guaranteed a subgraph on O(n + p n jPj jSj) edges that preserves the reachability of all pairs in P. We complement this with a lower bound graph construction, establishing that the above result fully characterizes the settings in which we are guaranteed a preserver of size O(n). Moreover, we design an efficient algorithm that can always compute a preserver of existentially optimal size. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0:6+ϵ) of Chlamatac, Dinitz, Kortsarz, and Laekhanukit (SODA'17) to O(n0:577+ϵ).