Reviving Thorup's Shortcut Conjecture

2025-10-28

AI Summary

Thorup’s reachability shortcut conjecture asked whether adding only $m^{1+o(1)}$ transitive closure edges to an $m$-edge graph $G$ could reduce its diameter to $m^{o(1)}$. Though refuted by Hesse (2003), we revive its core intuition in a generalized model permitting Steiner vertices. Method: We introduce a new shortcut framework allowing both Steiner vertices and shortcut edges, and develop generalized shortcut structures via path thickness analysis, explicit construction, and graph expansion techniques. Contribution/Results: Our structures simultaneously preserve approximate distances and flows—breaking all prior shortcut lower bounds. We provide candidate hard instances and derive a depth-$m^{o(1)}$ near-optimal parallel algorithm, achieving significant improvements over state-of-the-art methods for maximum flow and path queries.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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 $ ext{dist}_{G_+}(u,v) leq m^{o(1)}$ for all $(u,v)in E^*$, where $G_+=(V,Ecup E_+)$. 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.

Problem

• Investigating reachability shortcuts with Steiner vertices for graphs
• Breaking known shortcut lower bounds when allowing Steiner vertices
• Developing parallel algorithms for exact shortcut paths and maximum flow

Innovation

• Adding Steiner vertices to break shortcut lower bounds
• Ruling out ideal shortcuts with low thickness constraints
• Proposing candidate hard instances for revised conjecture