For many popular graph metric sparsifiers, such as spanners, emulators, and preservers, simple and elegant greedy algorithms are known that achieve state-of-the-art or existentially optimal tradeoffs between size and quality. The goal of this paper is to develop and analyze comparable greedy algorithms for nearby objects in graph metric augmentation. We show the following: - A simple greedy algorithm for shortcut sets achieves the state-of-the-art size/hopbound tradeoff recently proved by Kogan and Parter (2022), up to $O(\log n)$ factors in the size. Moreover, with an additional preprocessing step, the greedy algorithm subpolynomially improves on the previous size bounds in some range of parameters. - The same greedy algorithm was already known to be existentially optimal for the size/hopbound tradeoff for hopsets, by an analysis of Berman, Raskhodnikova, and Ruan (2010) introduced for transitive-closure spanners. We provide a completely different analysis showing that the algorithm is also existentially optimal (up to $O(\log n)$ factors) for the matching hopset problem, in which one has a budget of roughly $O(m)$ additional edges (for an $m$-edge input graph).

翻译：对于许多流行的图度量稀疏化结构（如展宽图、仿真图与保持图），已知简单而优雅的贪心算法能在规模与质量间实现当前最优或存在性最优的权衡。本文旨在为图度量增广中的近邻对象开发并分析类似的贪心算法。我们证明如下结果：- 针对捷径集的简单贪心算法能够实现Kogan与Parter（2022）近期证明的当前最优规模/跳数权衡（在规模上至多相差$O(\log n)$因子）。此外，通过增加预处理步骤，该贪心算法在部分参数范围内亚多项式地改进了先前规模上界。- 同一贪心算法已被Berman、Raskhodnikova与Ruan（2010）针对传递闭包展宽图的分析证明，在规模/跳数权衡上对跳跃集具有存在性最优性。我们提供了一种完全不同的分析，证明该算法在匹配跳跃集问题（其中给定预算约为$O(m)$条附加边，针对$m$条边的输入图）中也具有存在性最优性（至多相差$O(\log n)$因子）。