Single-Source Shortest Paths with Negative Real Weights in $\tilde{O}(mn^{8/9})$ Time (2311.02520v2)

Abstract: This paper presents a randomized algorithm for the problem of single-source shortest paths on directed graphs with real (both positive and negative) edge weights. Given an input graph with $n$ vertices and $m$ edges, the algorithm completes in $\tilde{O}(mn^{8/9})$ time with high probability. For real-weighted graphs, this result constitutes the first asymptotic improvement over the classic $O(mn)$-time algorithm variously attributed to Shimbel, BeLLMan, Ford, and Moore.

• The paper introduces a randomized algorithm that achieves a ~O(mn^(8/9)) time complexity for SSSP with negative real weights.
• It employs innovative techniques like betweenness reduction and hop-reduction to efficiently manage and reweight h-limited paths.
• The method concentrates computational resources on critical graph substructures, substantially reducing overhead compared to traditional approaches.

An Algorithm for Single-Source Shortest Paths with Negative Real Weights in Subquadratic Time

The paper introduces a randomized algorithm to address the Single-Source Shortest Paths (SSSP) problem in directed graphs, where edge weights can be both positive and negative real numbers. This proposed algorithm achieves a time complexity of $\tilde{O}(mn^{8/9})$ with high probability, an improvement over the $O(mn)$ run-time of the classical BeLLMan-Ford algorithm. This represents a significant advancement for graphs with real-weighted edges and marks the first asymptotic improvement in this domain.

Theoretical Foundation and Novel Contribution

The core contribution of this research is the development of a randomized approach that tackles the SSSP problem by introducing innovative techniques such as betweenness reduction and hop-reduction, enabling the processing of $h$-limited paths efficiently and effectively. The approach leverages properties of price functions, ensuring elimination of negative weights through a series of deterministic steps that include sampling and gradual reweighting of the graph.

Key Techniques Employed

1. Betweenness Reduction: This method reduces the number of vertices that lie on short paths between any two vertices in the graph. By reweighting the graph to ensure that low betweenness paths are non-negative, it becomes possible to focus computationally expensive steps on more critical parts of the graph.
2. Hop-Reduction Technique: By transposing $r$-remote structures in the graph into smaller paths, the algorithm efficiently executes over permutations of graph layers and mitigates the computational overhead associated with each path traversal. This technique ensures that the analysis of $h$-limited paths accounts for potential cycles and includes process optimization for finding negative edges only in crucial sections of the graph.
3. Finding Negative Sandwiches or Independent Sets: The algorithm either identifies a negative sandwich, a construct allowing for easy identification of paths needing correction, or a significant independent set of vertices. This aids in further localizing computation to significant substructures within the graph and enhances the algorithm's efficiency.

The Algorithm and Computational Complexity

The algorithm, while conceptually complex, efficiently reduces the problem size by focusing on subsets of negative edges and operating within bounded computational cycles. The decision to focus on $r$-remote sets in combination with hop-reduced paths implements a means to concentrate computational resources where they are most impactful, achieving a complexity of $\tilde{O}(mn^{8/9})$.

Implications and Future Directions

This breakthrough in handling negative weights in real-weighted graphs expands the algorithmic understanding of SSSP problems and sets precedence for further reducing complexity in dynamic and static network applications. The recognition of hop-minimized paths suggests potential adaptations for parallel processing or distributed computing environments. Furthermore, additional research could investigate deterministic variants and analyze the algorithm's performance on real-world data sets, specifically in networks where edge weights dynamically change in real-time applications like telecommunications and transportation.

In conclusion, this paper makes substantial progress in understanding and addressing the SSSP problem with negative weights, introducing novel techniques that significantly impact both theoretical computer science and practical applications. This reduces computational burden and enables closer examination and improved solutions to graph-theoretical problems that model complex, dynamic systems.