SIAM Journal on Computing, Volume 53, Issue 2, Page 221-246, April 2024.
Abstract. We define and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph [math] and a set of demand pairs [math], a reachability preserver is a sparse subgraph [math] 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 [math]-node graph and demand pairs of the form [math] for a small node subset [math], there is always a reachability preserver on [math] edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which [math] 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 [math] of Chlamtáč et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534–553] to [math].

中文翻译：

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可达性保护器：新的极值界限和近似算法

SIAM 计算杂志，第 53 卷，第 2 期，第 221-246 页，2024 年 4 月。
摘要。我们定义并研究可达性保护器，这是一种隐含在先前网络设计工作中的图论原语。给定一个有向图[数学]和一组需求对[数学]，可达性保护器是一个稀疏子图[数学]，它保留所有需求对之间的可达性我们的第一个贡献是可达性保护器大小的一系列极值界限。我们的主要结果表明，对于 [math] 节点图和小节点子集 [math] 形式的 [math] 需求对，[math] 边上始终存在可达性保护器。我们还给出了一个下界构造，证明该上限描述了在大范围参数中通常可能实现[数学]大小可达性保护器的设置。本文的第二个贡献是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，在这项工作之前，这两个领域之间的技术渗透一直是肤浅的。这使我们能够根据 Chlamtáč 等人的[数学]改进有向图中最基本的 Steiner 型问题的最先进的近似算法。 [近似扳手和有向斯坦纳森林：上限和下限，第二十八届年度 ACM-SIAM 离散算法研讨会论文集，SIAM，费城，2017 年，第 534-553 页] 至 [数学]。