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PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting Algorithms with Polynomially-Improved Approximation Factors for the $2 \rightarrow q$ Norm, and Applications A computational phase transition for learning-to-sample from Ising models Covering vertices by sequential stars Fermi-Dirac machines as quantizations of neurons A Comprehensive Evaluation of Vertex Elimination Algorithms for Algorithmic Differentiation A Tight Bound on Localization of Electrical Flows Optimal Dimension-Free Sampling for Regularized Classification Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs Beyond the Half-Approximation: Fair and Efficient Online Class Matching Efficient Uniform Sampling of Surjections via their Profiles Tractable Maximization of Budgeted Phylogenetic Diversity on Networks Utilizing Node Scanwidth Fairness in Aggregation: Optimal Top-$k$ and Improved Full Ranking Learning-Augmented Online Scheduling with Parsimonious Preemption Entropy Equivalence Testing Lumberjack: Better Differentially Private Random Forests through Heavy Hitter Detection in Trees The Secretary Problem with a Stochastic Precursor Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals Efficient Banzhaf-Based Data Valuation for $k$-Nearest Neighbors Classification Block-Sphere Vector Quantization An Approximation Algorithm for Graph Label Selection Iterative Chow Filtering for Learning with Distribution Shift Complexity of Non-Log-Concave Sampling in Fisher Information Stochastic Matching via Local Sparsification Finite Sample Bounds for Learning with Score Matching What is Learnable in Valiant's Theory of the Learnable? Provable Quantization with Randomized Hadamard Transform Min-Max Optimization Requires Exponentially Many Queries Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm A proximal gradient algorithm for composite log-concave sampling
Reviving Thorup's Shortcut Conjecture
Aaron Bernstein, Henry Fleischmann, Maximilian Probst Gutenberg, · 2025-10-29 · via cs.DS updates on arXiv.org

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.