Reviving Thorup's Shortcut Conjecture

TL;DR

This work revisits Thorup's Shortcut Conjecture by embracing Steiner vertices, reframing the goal as Steiner shortcuts that preserve reachability while achieving polylogarithmic diameter and near-linear size. It shows that the original lower-bound barriers evaporate when Steiner vertices are allowed, via explicit constructions that beat prior bounds, and introduces thickness as a central parameter governing shortcut viability. Nevertheless, the authors prove a negative result: even with Steiner vertices, ideal $m^{1+o(1)}$-size and $m^{o(1)}$-diameter shortcuts require thickness $t$ that grows beyond certain sublogarithmic bounds, and they propose a noised graph class as a harder instance. Beyond combinatorial bounds, the paper connects Steiner shortcuts to circuit-depth reductions and to parallel algorithms for exact shortest paths and maximum flow, arguing that near-optimal Steiner shortcuts could enable dramatically faster parallel solutions in directed reachability, exact SSSP, and max-flow. If the Steiner conjecture holds or the proposed hard instances are resolved, the work points toward near-linear-work, polylog-depth parallel algorithms for fundamental graph problems.

Abstract

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 $n^{1/2+o(1)}$ [Rozhoň, Haeupler, Martinsson, STOC'23] and $m^{1+o(1)}$ [Chen, Kyng, Liu, FOCS'22], respectively.

Theorems & Definitions (59)

• conjecture 1: Thorup's Shortcut Conjecture thorup1993shortcutting
• theorem 1.1: Hesse Hesse03
• conjecture 2: Steiner Shortcut Conjecture
• definition 1.2: Thickness
• Theorem 1.3
• conjecture 3
• theorem 2.1: Cf. Hesse03
• theorem 2.2
• remark 2.3
• lemma 2.4: Cf. HuangP21