We abstract and study \emph{reachability preservers}, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph $G = (V, E)$ and a set of \emph{demand pairs} $P \subseteq V \times V$, a reachability preserver is a sparse subgraph $H$ that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an $n$-node graph and demand pairs of the form $P \subseteq S \times V$ for a small node subset $S$, there is always a reachability preserver on $O(n+\sqrt{n |P| |S|})$ edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which $O(n)$ size reachability preservers are generally possible, in a large range of parameters. 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 $O(n^{0.6+\varepsilon})$ of Chlamatac, Dinitz, Kortsarz, and Laekhanukit (SODA'17) to $O(n^{4/7+\varepsilon})$.

翻译：我们抽象并研究了\emph{可达性保持器}，这是一种在图论网络设计中隐含在先期工作中的基本结构。给定有向图$G = (V, E)$和一组\emph{需求对}$P \subseteq V \times V$，可达性保持器是一个稀疏子图$H$，它保留了所有需求对之间的可达性。我们的第一个贡献是一系列关于可达性保持器规模的极值界限。主要结果表明，对于$n$个节点的图以及形式为$P \subseteq S \times V$（其中$S$为小型节点子集）的需求对，总是存在边数为$O(n+\sqrt{n |P| |S|})$的可达性保持器。此外，我们给出了一个下界构造，证明该上界刻画了在大范围参数下通常能实现$O(n)$规模可达性保持器的条件。本文的第二个贡献是揭示了极值图稀疏化结果与经典斯坦纳网络设计问题之间的新联系。令人惊讶的是，在此工作之前，这两个领域之间的技术渗透仅是表面性的。这使得我们能够将Chlamatac、Dinitz、Kortsarz和Laekhanukit（SODA'17）提出的有向图中最基本斯坦纳型问题的近似算法从$O(n^{0.6+\varepsilon})$改进至$O(n^{4/7+\varepsilon})$。