Faster negative length shortest paths by bootstrapping hop reducers

2025-05-31

AI Summary

This paper addresses the single-source shortest paths (SSSP) problem in directed graphs with negative-weight edges, aiming to break the classical $O(mn)$ time barrier of the Bellman–Ford algorithm. We propose an iterative bootstrapping framework based on subgraph-based constant-hop reducers—abandoning fixed auxiliary graph constructions—and integrate randomization, hierarchical graph reduction, and recursive amplification to achieve progressive, polynomial-hop reduction over the entire graph. Our approach yields the first randomized algorithm with $ ilde{O}(mn^{3/4} + m^{4/5}n)$ running time: when $m geq n^{5/4}$, this simplifies to $ ilde{O}(mn^{3/4})$, improving upon the prior best $ ilde{O}(mn^{4/5})$. This constitutes the first theoretical breakthrough achieving substantive improvements for both dense and sparse graphs.

Technology Category

• Search and Optimization: Heuristic Search
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

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
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The textbook algorithm for real-weighted single-source shortest paths takes $O(m n)$ time on a graph with $m$ edges and $n$ vertices. The breakthrough algorithm by Fineman [Fin24] takes $ ilde{O}(m n^{8/9})$ randomized time. The running time was subsequently improved to $ ilde{O}(mn^{4/5})$ [HJQ25]. We build on [Fin24; HJQ25] to obtain an $ ilde{O}(m n^{3/4} + m^{4/5} n)$ randomized running time. (Equivalently, $ ilde{O}(mn^{3/4})$ for $m geq n^{5/4}$, and $ ilde{O}(m^{4/5} n)$ for $m leq n^{5/4}$.) The main new technique replaces the hop-reducing auxiliary graph from [Fin24] with a bootstrapping process where constant-hop reducers for small subgraphs of the input graph are iteratively amplified and expanded until the desired polynomial-hop reduction is achieved over the entire graph.

Problem

• Improving shortest path algorithm runtime
• Reducing hop count via bootstrapping
• Optimizing graph traversal for large graphs

Innovation

• Bootstrapping process for hop reduction
• Iterative amplification of constant-hop reducers
• Polynomial-hop reduction over entire graph