Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness

2026-02-11

AI Summary

This work investigates how to accelerate parallel connectivity and shortest-path algorithms by efficiently augmenting a directed acyclic graph with transitive-closure edges to substantially reduce its diameter. The authors introduce a structural criterion called “certified shortcut edges,” which precisely characterizes the set of shortcut edges constructible by any near-linear-time algorithm. Leveraging this criterion in conjunction with combinatorial graph theory and complexity lower-bound techniques, they improve the known diameter lower bound from $n^{2/9 - o(1)}$ to $n^{1/4 - o(1)}$, establishing that no near-linear-time algorithm can achieve a smaller diameter. This result significantly strengthens existing theoretical limits and provides a robust foundation for the design of future efficient shortcut-edge construction methods.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

All parallel algorithms for directed connectivity and shortest paths crucially rely on efficient shortcut constructions that add a linear number of transitive closure edges to a given DAG to reduce its diameter. A long sequence of works has studied both (efficient) shortcut constructions and impossibility results on the best diameter and therefore the best parallelism that can be achieved with this approach. This paper introduces a new conceptual and technical tool, called certified shortcuts, for this line of research in the form of a simple and natural structural criterion that holds for any shortcut constructed by an efficient (combinatorial) algorithm. It allows us to drastically simplify and strengthen existing impossibility results by proving that any near-linear-time shortcut-based algorithm cannot reduce a graph's diameter below $n^{1/4-o(1)}$. This greatly improves over the $n^{2/9-o(1)}$ lower bound of [HXX25] and seems to be the best bound one can hope for with current techniques. Our structural criterion also precisely captures the constructiveness of all known shortcut constructions: we show that existing constructions satisfy the criterion if and only if they have known efficient algorithms. We believe our new criterion and perspective of looking for certified shortcuts can provide crucial guidance for designing efficient shortcut constructions in the future.

Problem

• shortcuts
• diameter bounds
• directed connectivity
• parallel algorithms
• constructiveness

Innovation

• certified shortcuts
• diameter lower bound
• parallel algorithms
• graph shortcuts
• constructiveness criterion