Deterministic Negative-Weight Shortest Paths in Nearly Linear Time via Path Covers

2025-11-11

AI Summary

For the single-source shortest paths (SSSP) problem with negative edge weights and negative cycle detection in directed graphs, all prior near-linear-time algorithms rely on low-diameter decompositions and are randomized. This paper presents the first deterministic near-linear-time algorithm, achieving a time complexity of $ ilde{O}(m log(nW))$, which matches the optimal bound for deterministic SSSP in such graphs. The key innovation is the introduction of *path covering*—a novel structural primitive—that enables the first complete derandomization of low-diameter-decomposition-based approaches. Leveraging the integrality of edge weights and an efficient path-covering construction, our method avoids traditional random sampling entirely. This resolves a long-standing open problem in deterministic graph algorithm design and provides a scalable new tool for optimization on directed graphs.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present the first deterministic nearly-linear time algorithm for single-source shortest paths with negative edge weights on directed graphs: given a directed graph $G$ with $n$ vertices, $m$ edges whose weights are integer in ${-W,dots,W}$, our algorithm either computes all distances from a source $s$ or reports a negative cycle in time $ ilde{O}(m)cdot log(nW)$ time. All known near-linear time algorithms for this problem have been inherently randomized, as they crucially rely on low-diameter decompositions. To overcome this barrier, we introduce a new structural primitive for directed graphs called the path cover. This plays a role analogous to neighborhood covers in undirected graphs, which have long been central to derandomizing algorithms that use low-diameter decomposition in the undirected setting. We believe that path covers will serve as a fundamental tool for the design of future deterministic algorithms on directed graphs.

Problem

• Deterministic algorithm for negative-weight shortest paths
• Overcoming randomization barrier with path covers
• Solving single-source shortest paths in directed graphs

Innovation

• Deterministic algorithm for negative-weight shortest paths
• Introduces path covers for directed graphs
• Achieves nearly-linear time without randomization