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的边 $H$，当将其添加到 $G$ 中时，保持其图度量，但保证所有节点对在 $G \cup H$ 中都有至多使用 $D$ 条边的最短路径。捷径集是可达性的类似概念。自 20 世纪 90 年代初以来，这些对象因其在并行、分布式、动态和流图算法中的应用而受到研究。在它们的大部分历史中，这两种对象的最先进构造都是基于随机采样节点以击中图中长路径的简单民间算法。然而，Kogan 和 Parter [SODA '22] 以及 Bernstein 和 Wein [SODA '23] 的最新突破最终改进了捷径集和 $(1+\epsilon)$-近似跳集的民间直径界 $\widetilde{O}(n^{1/2})$。对于这两种对象，现在已知可以使用 $O(n)$ 跳边将直径减少到 $\widetilde{O}(n^{1/3})$。民间采样仍未得到改进的唯一场景是精确跳集。这些改进能否继续？我们通过构造图来否定地回答这个问题，在这些图上，任何 $O(n)$ 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] 先前的下界 $\widetilde{\Omega}(n^{1/3})$。使用类似的思想，我们还多项式地改进了当前捷径集的下界，构造了图，在这些图上任何 $O(n)$ 条边的捷径集将直径减少到 $\widetilde{\Omega}(n^{1/4})$。这改进了 Huang 和 Pettie [SIAM J. Disc. Math. '18] 先前的下界 $\Omega(n^{1/6})$。我们还扩展了我们的构造，为 $O(p)$ 大小的精确跳集和捷径集提供了其他 $p$ 值的下界；特别地，我们表明在 $p \in [1, n^2]$ 的整个范围内，民间采样对于精确跳集几乎是