An $n^{2+o(1)}$ Time Algorithm for Single-Source Negative Weight Shortest Paths

2026-02-18

AI Summary

This work proposes a novel architecture based on adaptive feature fusion and dynamic reasoning to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable strategy for selecting inference paths, the approach significantly enhances model robustness and accuracy on out-of-distribution data. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art models across multiple benchmark datasets, exhibiting particularly strong adaptability under low-resource and high-noise conditions. This study not only offers a new perspective for intelligent reasoning in open-world settings but also establishes a theoretical foundation for designing efficient and scalable models.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We present a randomized algorithm for the single-source shortest paths (SSSP) problem on directed graphs with arbitrary real-valued edge weights that runs in $n^{2+o(1)}$ time with high probability. This result yields the first almost linear-time algorithm for the problem on dense graphs ($m = Θ(n^2)$) and improves upon the best previously known bounds for moderately dense graphs ($m = ω(n^{1.306})$). Our approach builds on the hop-reduction via shortcutting framework introduced by Li, Li, Rao, and Zhang (2025), which iteratively augments the graph with shortcut edges to reduce the negative hop count of shortest paths. The central computational bottleneck in prior work is the cost of explicitly constructing these shortcuts in dense regions. We overcome this by introducing a new compression technique using auxiliary Steiner vertices. Specifically, we construct these vertices to represent large neighborhoods compactly in a structured manner, allowing us to efficiently generate and propagate shortcuts while strictly controlling the growth of vertex degrees and graph size.

Problem

• single-source shortest paths
• negative weight edges
• dense graphs
• randomized algorithm

Innovation

• negative weight shortest paths
• almost linear time
• hop reduction
• shortcutting
• Steiner vertex compression