Simpler and Higher Lower Bounds for Shortcut Sets

Abstract

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 $\widetildeΩ(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set problem. We highlight that our simplification completely removes the use of the convex sets by Bárány 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 $Ω(n^{1/5})$ lower bound, improving on the previous best $Ω(n^{1/8})$ lower bound. For all $\varepsilon > 0$, we show that there exists a $δ> 0$ 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 $Ω(n^δ)$. 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 $|S| = \widetildeΘ(n^{3/(d+3)})$, such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is $\widetildeΩ(|S|^{1/3})$.