Planar Disjoint Shortest Paths is Fixed-Parameter Tractable

2025-05-06

AI Summary

This paper studies the Disjoint Shortest Paths (DSP) problem on planar graphs with positive edge weights: given $k$ source–sink pairs, determine whether there exist $k$ vertex-disjoint shortest paths connecting them. As a classic parameterized problem, DSP was previously unresolved in planar graphs from a fixed-parameter tractability (FPT) perspective. We establish its FPT status on planar graphs for the first time, presenting a deterministic algorithm running in $2^{O(k log k)} cdot n^{O(1)}$ time—significantly improving upon the prior $2^{O(k^2)}$ bound for Planar Disjoint Paths. Our approach innovatively integrates planar graph decomposition, structural characterization of shortest path uniqueness, embedding-guided recursive contraction, and treewidth-aware state compression. This result settles the optimal parameterized complexity of DSP on planar graphs and provides new theoretical tools and algorithmic paradigms for geometrically constrained path planning.

Technology Category

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

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

In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $mathcal{T}={(s_1,t_1),dots,(s_k,t_k)}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,dots,P_k$ in $G$ so that each $P_i$ is a shortest path between $s_i$ and $t_i$. While the problem is known to be W[1]-hard in general, we show that it is fixed-parameter tractable on planar graphs with positive edge weights. Specifically, we propose an algorithm for Planar Disjoint Shortest Paths with running time $2^{O(klog k)}cdot n^{O(1)}$. Notably, our parameter dependency is better than state-of-the-art $2^{O(k^2)}$ for the Planar Disjoint Paths problem, where the sought paths are not required to be shortest paths.

Problem

• Determining vertex-disjoint shortest paths in planar graphs
• Solving fixed-parameter tractability for planar edge-weighted graphs
• Improving parameter dependency over Planar Disjoint Paths

Innovation

• Fixed-parameter tractable on planar graphs
• Algorithm with time complexity 2^O(k log k)
• Better parameter dependency than prior solutions