We aim to revive Thorup's conjecture [Thorup, WG'92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph $G=(V,E)$ with $m$ edges, we can add $m^{1+o(1)}$ ``shortcut'' edges $E_+$ from the transitive closure $E^*$ of $G$ so that $\text{dist}_{G_+}(u,v) \leq m^{o(1)}$ for all $(u,v)\in E^*$, where $G_+=(V,E\cup E_+)$. The conjecture was refuted by Hesse [Hesse, SODA'03], followed by significant efforts in the last few years to optimize the lower bounds. In this paper we observe that although Hesse refuted the letter of Thorup's conjecture, his work~[Hesse, SODA'03] -- and all followup work -- does not refute the spirit of the conjecture, which should allow $G_+$ to contain both new (shortcut) edges and new Steiner vertices. Our results are as follows. (1) On the positive side, we present explicit attacks that break all known shortcut lower bounds when Steiner vertices are allowed. (2) On the negative side, we rule out ideal $m^{1+o(1)}$-size, $m^{o(1)}$-diameter shortcuts whose ``thickness'' is $t=o(\log n/\log \log n)$, meaning no path can contain $t$ consecutive Steiner vertices. (3) We propose a candidate hard instance as the next step toward resolving the revised version of Thorup's conjecture. Finally, we show promising implications. Almost-optimal parallel algorithms for computing a generalization of the shortcut that approximately preserves distances or flows imply almost-optimal parallel algorithms with $m^{o(1)}$ depth for exact shortcut paths and exact maximum flow. The state-of-the-art algorithms have much worse depth of $n^{1/2+o(1)}$ [Rozho\v{n}, Haeupler, Martinsson, STOC'23] and $m^{1+o(1)}$ [Chen, Kyng, Liu, FOCS'22], respectively.

翻译：本文旨在复兴Thorup关于存在具有理想尺寸-直径权衡的可达性捷径的猜想[Thorup, WG'92]。Thorup最初提出：给定任意具有m条边的图G=(V,E)，我们能否从G的传递闭包E*中添加m^{1+o(1)}条“捷径”边E_+，使得对于所有(u,v)∈E*，满足dist_{G_+}(u,v) ≤ m^{o(1)}，其中G_+=(V,E∪E_+)。该猜想被Hesse证伪[Hesse, SODA'03]，随后近些年出现了大量优化下界的研究工作。本文指出，尽管Hesse证伪了Thorup猜想字面表述，但其工作[Hesse, SODA'03]——以及所有后续研究——并未证伪该猜想的精神实质，即应允许G_+同时包含新（捷径）边和新Steiner顶点。我们的研究结果如下：（1）在正面结果方面，我们提出了显式攻击方法，当允许Steiner顶点时，可突破所有已知的捷径下界。（2）在负面结果方面，我们排除了厚度为t=o(log n/log log n)的理想m^{1+o(1)}尺寸、m^{o(1)}直径捷径的存在性，其中“厚度”指任意路径不能包含t个连续的Steiner顶点。（3）我们提出了一个候选困难实例，作为解决Thorup猜想修订版的下一步工作。最后，我们展示了有前景的应用意义：计算近似保持距离或流的捷径泛化模型的近乎最优并行算法，意味着对于精确捷径路径和精确最大流问题，可得到具有m^{o(1)}深度的近乎最优并行算法。当前最先进算法的深度分别为n^{1/2+o(1)} [Rozhoň, Haeupler, Martinsson, STOC'23]和m^{1+o(1)} [Chen, Kyng, Liu, FOCS'22]，远逊于此。