We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of the graph. Our results are based on a vast simplification of the recent construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an $\widetilde{\Omega}(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set problem. We highlight that our simplification completely removes the use of the convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous lower bound constructions. Our simplification also removes the need for randomness and further removes some log factors. This allows us to generalize the construction to higher dimensions, which in turn can be used to show the following results. For $O(m)$-sized shortcut sets, we show an $\Omega(n^{1/5})$ lower bound, improving on the previous best $\Omega(n^{1/8})$ lower bound. For all $\varepsilon > 0$, we show that there exists a $\delta > 0$ such that there are $n$-vertex $O(n)$-edge graphs $G$ where adding any shortcut set of size $O(n^{2-\varepsilon})$ keeps the diameter of $G$ at $\Omega(n^\delta)$. This improves the sparsity of the constructed graph compared to a known similar result by Hesse [SODA 2003]. We also consider the sourcewise setting for shortcut sets: given a graph $G=(V,E)$, a set $S\subseteq V$, how much can we decrease the sourcewise diameter of $G$, $\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty} \text{dist}(s,v)$ by adding a set of edges $H$ from the transitive closure of $G$? We show that for any integer $d \ge 2$, there exists a graph $G=(V, E)$ on $n$ vertices and $S \subseteq V$ with $|S| = \widetilde{\Theta}(n^{3/(d+3)})$, such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is $\widetilde{\Omega}(|S|^{1/3})$.

翻译：我们为著名的捷径集问题提供了多种下界：通过从有向图的传递闭包中添加$O(n)$或$O(m)$条捷径，能在多大程度上减小包含$n$个顶点和$m$条边的有向图的直径？我们的结果基于对Bodwin和Hoppenworth [FOCS 2023]最新构造的极大简化，该构造曾用于证明$O(n)$大小的捷径集问题的$\widetilde{\Omega}(n^{1/4})$下界。我们强调，我们的简化完全去除了之前所有下界构造中使用的Bárány和Larman [Math. Ann. 1998]的凸集方法。该简化还消除了对随机性的需求，并进一步移除了一些对数因子。这使我们能够将构造推广到更高维度，进而可用于证明以下结果。对于$O(m)$大小的捷径集，我们证明了$\Omega(n^{1/5})$的下界，改进了先前最佳的$\Omega(n^{1/8})$下界。对于所有$\varepsilon > 0$，我们证明存在$\delta > 0$，使得存在包含$n$个顶点和$O(n)$条边的图$G$，添加任意大小为$O(n^{2-\varepsilon})$的捷径集后，$G$的直径仍保持在$\Omega(n^\delta)$。这相较于Hesse [SODA 2003]已知的类似结果，改进了所构造图的稀疏性。我们还考虑了捷径集的源点设置：给定图$G=(V,E)$和顶点子集$S\subseteq V$，通过从$G$的传递闭包中添加边集$H$，能在多大程度上减小$G$的源点直径$\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty} \text{dist}(s,v)$？我们证明，对任意整数$d \ge 2$，存在包含$n$个顶点的图$G=(V, E)$和满足$|S| = \widetilde{\Theta}(n^{3/(d+3)})$的$S \subseteq V$，使得当添加$O(n)$或$O(m)$条捷径时，源点直径为$\widetilde{\Omega}(|S|^{1/3})$。