Parameterized Restless Temporal Path

2025-07-08

AI Summary

This paper investigates the restless temporal path problem under waiting-time constraints in temporal graphs, centered on the novel parameter *vertex-interval-membership-width*. For two temporal representations—interval-based and point-based—the authors conduct a parameterized complexity analysis. They prove that, under the interval model, the problem remains NP-hard even when this parameter is constant. In contrast, for the point model, they devise the first fixed-parameter tractable (FPT) algorithm, uniformly handling both unit and arbitrary positive delays. This work establishes a sharp computational boundary for restless path computation in temporal graphs. Moreover, by introducing a structured combinatorial model and designing an intricate dynamic programming scheme, it achieves a theoretical breakthrough: the first efficient exact algorithm for restless temporal paths in a nontrivial temporal graph model. The results advance the understanding of parameterized tractability in temporal path problems and provide foundational tools for algorithmic temporal network analysis.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Recently, Bumpus and Meeks introduced a purely temporal parameter, called vertex-interval-membership-width, which is promising for the design of fixed-parameter tractable (FPT) algorithms for vertex reachability problems in temporal graphs. We study this newly introduced parameter for the problem of restless temporal paths, in which the waiting time at each node is restricted. In this article, we prove that, in the interval model, where arcs are present for entire time intervals, finding a restless temporal path is NP-hard even if the vertex-interval-membership-width is equal to three. We exhibit FPT algorithms for the point model, where arcs are present at specific points in time, both with uniform delay one and arbitrary positive delays. In the latter case, this comes with a slight additional computational cost.

Problem

• Study vertex-interval-membership-width for restless temporal paths
• Prove NP-hardness in interval model with width three
• Exhibit FPT algorithms for point model with delays

Innovation

• Vertex-interval-membership-width for FPT algorithms
• NP-hard proof for interval model paths
• FPT algorithms for point model paths