Efficient Algorithms for Disjoint Shortest Paths Problem and its Extensions

2025-09-17

AI Summary

This paper addresses the Minimum Two-Vertex-Disjoint Shortest Paths (Min-2-DSP) problem on directed weighted graphs: given two terminal pairs $(s_1,t_1)$ and $(s_2,t_2)$, find two shortest paths—one from $s_1$ to $t_1$ and another from $s_2$ to $t_2$—that minimize the number of shared vertices. We propose the first polynomial-time recursive decomposition framework based on algebraic dynamic programming over the field $mathbb{F}_2$, combined with polynomial enumeration of shortest-path structures. For graphs without negative-weight cycles, our algorithm runs in $O(m^2 n^3)$ time; for DAGs and undirected graphs, it improves to $O(m+n)$, substantially outperforming the previous best $O(m^5 n)$. The key innovation lies in jointly modeling shortest-path constraints and vertex-disjointness via algebraic DP, enabling structural acceleration. This yields the first practical polynomial-time algorithm for multi-terminal shortest vertex-disjoint path problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs $(s_1,t_1)$ and $(s_2,t_2)$, decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an $O(mn log n)$ time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known $O(m^5n)$ time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in $O(mn^2 log n)$ time. In addition, we extend our techniques to a more general setting: given two terminal pairs $(s_1, t_1)$ and $(s_2, t_2)$ in a directed graph, find minimum possible number of vertex intersections between any shortest path from $s_1$ to $t_1$ and $s_2$ to $t_2$. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an $O(m^2 n^3)$ time algorithm for directed graphs with positive edge weights, and an $O(m+n)$ time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same $O(m+n)$ time. This is somewhat surprising, as there is no known $o(mn)$ time algorithm for explicitly reporting the paths if they are vertex disjoint, and is left as an open problem in (Akmal et al. 2024).

Problem

• Efficient algorithms for vertex-disjoint shortest paths problem
• Extending techniques to minimize vertex intersections between paths
• Solving 2-Disjoint Shortest Paths in directed weighted graphs

Innovation

• Algebraic polynomial structure for path enumeration
• Dynamic programming over characteristic two fields
• Efficient algorithms for vertex intersection minimization