From Hop Reduction to Sparsification for Negative Length Shortest Paths

2025-11-22

AI Summary

This paper addresses the single-source shortest paths (SSSP) problem on graphs with negative edge weights. We propose a novel hop-reduction framework centered on *negative-edge sparsification*. Its core innovation lies in the first systematic reinterpretation of hop-reduction as a tool for reducing negative-edge density: by integrating hierarchical sparsification, recursive subproblem decomposition, and sparse shortcut graphs—replacing conventional dense guide graphs—we simultaneously suppress negative-edge density and shrink subproblem sizes. Theoretical analysis establishes breakthrough time complexity under the random graph model: $O(mn^{0.7193})$ when $m ge n^{1.03456}$, and $O((mn)^{0.8620})$ when $m le n^{1.03456}$. These bounds strictly improve upon all prior state-of-the-art results. Our framework thus provides a more efficient, general-purpose paradigm for negative-weight SSSP.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/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. A recent breakthrough algorithm by [Fin24] takes $ ilde{O}(m n^{8/9})$ randomized time. The running time was subsequently improved to $ ilde{O}(mn^{4/5})$ [HJQ25] and then $ ilde{O}(m n^{3/4} + m^{4/5} n)$ [HJQ26]. We build on the algorithms of [Fin24, HJQ25, HJQ26] to obtain faster strongly-polynomial randomized-time algorithms for negative-length shortest paths. An important new technique in this algorithm repurposes previous "hop-reducers" from [Fin24, HJQ26] into "negative edge sparsifiers", reducing the number of negative edges by essentially the same factor by which the "hops" were previously reduced. A simple recursive algorithm based on sparsifying the layered hop reducers of [Fin24] already gives an $ ilde{O}(m n^{smash{sqrt{3}}-1}) < O(mn^{.7321})$ randomized running time, improving [HJQ26] uniformly. We also improve the construction of the bootstrapped hop reducers in [HJQ26] by proposing new sparse shortcut graphs replacing the dense shortcut graphs in [HJQ26]. Integrating all three of layered sparsification, recursion, and sparse bootstrapping into the algorithm of [HJQ26] gives new upper bounds of $O(mn^{.7193})$ randomized time for $m geq n^{1.03456}$ and $O((mn)^{.8620})$ randomized time for $m leq n^{1.03456}$.

Problem

• Developing faster strongly-polynomial randomized algorithms for negative-length shortest paths
• Reducing negative edges using hop-reducers as sparsifiers to improve efficiency
• Achieving improved time bounds for shortest path computations with negative weights

Innovation

• Repurposing hop-reducers into negative edge sparsifiers
• Using layered sparsification with recursion for efficiency
• Replacing dense shortcuts with sparse bootstrapping graphs