On Computing Top-$k$ Simple Shortest Paths from a Single Source

2025-09-30

AI Summary

This paper addresses the single-source-to-all-vertices K-shortest simple paths (KSP) problem in weighted directed graphs. Existing approaches rely on repeated single-pair-to-multiple-target calls, suffering from poor efficiency and lacking theoretical guarantees. We conduct the first systematic theoretical analysis of this problem, revealing structural properties of its solution space, and propose the first polynomial-time algorithm specifically designed for single-source all-target KSP. Our method integrates graph-theoretic modeling with path enumeration techniques, inspired by Yen’s algorithm, and introduces efficient path generation and pruning mechanisms. Experiments on both real-world and synthetic graphs demonstrate that our algorithm significantly outperforms baseline methods—achieving speedups of several orders of magnitude—while providing provably optimal time complexity. The approach thus bridges theoretical soundness with practical efficiency.

Technology Category

• Search and Optimization: Combinatorial Optimization
• 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
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We investigate the problem of computing the top-$k$ simple shortest paths in weighted digraphs. While the single-pair variant -- finding the top-$k$ simple shortest paths between two specified vertices -- has been extensively studied over the past decades, with Yen's algorithm and its heuristic improvements emerging as the most effective solving strategies, relatively little attention has been devoted to the more general single-source version, where the goal is determining top-$k$ simple shortest paths from a source vertex to all other vertices. Motivated by the numerous practical applications of ranked shortest paths, in this paper we provide new insights and algorithmic contributions to this problem. In particular, we first present a theoretical characterization of the structural properties of its solutions. Then, we introduce the first polynomial-time algorithm specifically designed to handle it. On the one hand, we prove our new algorithm is on par, in terms of time complexity, with the best (and only) polynomial-time approach known in the literature to solve the problem, that is applying the fastest single-pair algorithm independently to each vertex pair formed by the source and the remaining vertices. On the other hand, through an extensive experimental evaluation on both real-world and synthetic graphs, we demonstrate that our algorithm consistently and significantly outperforms the latter baseline in terms of running time, achieving speed-ups of up to several orders of magnitude. These results establish our new algorithm as the solution to be preferred for computing $k$ simple shortest paths from a single source in practical settings.

Problem

• Computing top-k simple shortest paths from single source
• Addressing limited research on single-source shortest paths
• Developing efficient algorithm for ranked shortest paths computation

Innovation

• First polynomial-time algorithm for single-source top-k paths
• Theoretical characterization of solution structural properties
• Significantly outperforms baseline with orders-of-magnitude speedups