A Bottom-Up Algorithm for Negative-Weight SSSP with Integrated Negative Cycle Finding
Jason Li, Connor Mowry
TL;DR
This work addresses Negative-Weight SSSP with possible negative cycles by introducing a bottom-up, diameter-based decomposition (Scale) that integrates negative-cycle finding into a Bellman-Ford/Dijkstra hybrid. The method scales by first forming a scaled graph $G'$ with $w_{G'}(e)=w_G(e)+\frac{W}{2}$, recursively decomposing via a diameter parameter, and then computing distances iteratively while adjusting a potential $\phi$ to ensure non-negative edges, terminating early if a negative cycle is detected. The main contribution is a simpler, robust algorithm that achieves $O\left( (m + \log\log n) \log^2 n \right)$ expected time (high probability) and yields a certificate $G_\phi$ or a negative cycle, improving accessibility and implementation practicality while preserving correctness guarantees. The approach emphasizes robust parameterization and design simplicity, offering a practical alternative for solving Negative-Weight SSSP in graphs with potential negative cycles.
Abstract
We present a simplified algorithm for solving the Negative-Weight Single-Source Shortest Paths (SSSP) problem, focusing on enhancing clarity and practicality over prior methods. Our algorithm uses graph diameter as a recursive parameter, offering greater robustness to the properties of the decomposed graph compared to earlier approaches. Additionally, we fully integrate negative-weight cycle finding into the algorithm by augmenting the Bellman-Ford/Dijkstra hybrid, eliminating the need for a separate cycle-finding procedure found in prior methods. Although the algorithm achieves no theoretical efficiency gains, it simplifies negative cycle finding and emphasizes design simplicity, making it more accessible for implementation and analysis. This work highlights the importance of robust parameterization and algorithmic simplicity in addressing the challenges of Negative-Weight SSSP.