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$G$，直径为$D$的精确跳集是一小部分额外边集$H$，当加入$G$后，保持其图度量但保证所有节点对在$G \cup H$中存在一条至多$D$条边的最短路径。 shortcut集是可达性概念下的类似对象。自20世纪90年代初起，这些对象因在并行、分布式、动态和流式图算法中的应用而被研究。在其研究历史的大部分时间中，这两类对象的最先进构造均基于一个简单的民间算法——通过随机采样节点覆盖图中的长路径。然而，Kogan与Parter [SODA '22] 以及 Bernstein与Wein [SODA '23] 的最新突破，最终改进了shortcut集和$(1+\epsilon)$-近似跳集的民间直径界$\widetilde{O}(n^{1/2})$。目前已知，对于这两类对象，可使用$O(n)$条跳边将直径降至$\widetilde{O}(n^{1/3})$。唯一未被改进的民间采样应用场景是精确跳集。这些改进能否延续？我们通过构造图给出了否定回答：对于任何具有$O(n)$条边的精确跳集，其直径必为$\widetilde{\Omega}(n^{1/2})$。这改进了Kogan与Parter [FOCS '22] 此前得到的$\widetilde{\Omega}(n^{1/3})$下界。利用类似思路，我们还多项式地改进了shortcut集的现有下界：构造的图上任何具有$O(n)$条边的shortcut集，其直径降至$\widetilde{\Omega}(n^{1/4})$，这改进了Huang与Pettie [SIAM J. Disc. Math. '18] 的$\Omega(n^{1/6})$下界。我们进一步将构造推广至其他参数$p$的$O(p)$-规模精确跳集与shortcut集的下界；特别地，我们证明在$p \in [1, n^2]$的整个范围内，民间采样对于精确跳集具有近乎最优性。