For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$.

翻译：对于图$G$，一个$D$直径缩减精确跳集是一组额外边$H$，当添加到$G$中时，保持其图度量，但保证$G \cup H$中所有节点对的最短路径最多使用$D$条边。捷径集是可达性的类似概念。自90年代初以来，由于在并行、分布式、动态和流式图算法中的应用，这些对象一直受到研究。在其大部分历史中，这两种对象的最先进构造是一种简单的民间传说算法，基于随机采样节点以击中图中的长路径。然而，Kogan和Parter [SODA '22]以及Bernstein和Wein [SODA '23]的最新突破终于改进了捷径集和$(1+\epsilon)$近似跳集的民间传说直径界$\widetilde{O}(n^{1/2})$。对于这两种对象，现在已知可以使用$O(n)$跳边将直径减小到$\widetilde{O}(n^{1/3})$。唯一未改进的民间传说采样设置是精确跳集。这些改进能否持续？我们通过构造图来否定地回答这个问题，在这些图中，任何包含$O(n)$条边的精确跳集都有直径$\widetilde{\Omega}(n^{1/2})$。这改进了Kogan和Parter [FOCS '22]之前的$\widetilde{\Omega}(n^{1/3})$下界。使用类似思想，我们还多项式改进了当前捷径集的下界，构造图使得任何包含$O(n)$条边的捷径集将直径减小到$\widetilde{\Omega}(n^{1/4})$。这改进了Huang和Pettie [SIAM J. Disc. Math. '18]之前的$\Omega(n^{1/6})$下界。我们还扩展了构造，为$O(p)$大小的精确跳集和捷径集在其他$p$值下提供下界；特别地，我们证明在$p \in [1, n^2]$的整个范围内，民间传说采样对于精确跳集是近乎最优的。