Beer Path Problems in Temporal Graphs

2025-07-11

AI Summary

This paper addresses path planning in temporal graphs with time-sensitive points of interest (e.g., gas stations open during specific hours), introducing the “beer path” concept to the temporal graph setting for the first time. We formalize four variants: earliest-arrival, latest-departure, shortest-duration, and fewest-hop temporal beer paths. Methodologically, we propose an efficient algorithm leveraging dual graph representations—edge streams and adjacency lists—integrated with precomputation and dynamic graph transformation techniques to support queries under node activity changes (e.g., store openings/closures). Our algorithm achieves time complexity comparable to classical temporal shortest-path algorithms. Experiments demonstrate its rapid response capability in dynamic environments. The main contributions are: (i) establishing the first theoretical framework for temporal beer paths; (ii) proposing a unified, scalable, efficient, and adaptive solution paradigm; and (iii) enabling practical deployment in real-world dynamic temporal networks.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

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
• 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

Abstract

Computing paths in graph structures is a fundamental operation in a wide range of applications, from transportation networks to data analysis. The beer path problem, which captures the option of visiting points of interest, such as gas stations or convenience stops, prior to reaching the final destination, has been recently introduced and extensively studied in static graphs. However, existing approaches do not account for temporal information, which is often crucial in real-world scenarios. For instance, transit services may follow fixed schedules, and shops may only be accessible during certain hours. In this work, we introduce the notion of beer paths in temporal graphs, where edges are time-dependent and certain vertices (beer vertices) are active only at specific time instances. We formally define the problems of computing earliest-arrival, latest-departure, fastest, and shortest temporal beer paths and propose efficient algorithms for these problems under both edge stream and adjacency list representations. The time complexity of each of our algorithms is aligned with that of corresponding temporal pathfinding algorithms, thus preserving efficiency. Additionally, we present preprocessing techniques that enable efficient query answering under dynamic conditions, for example new openings or closings of shops. We achieve this through appropriate precomputation of selected paths or by transforming a temporal graph into an equivalent static graph.

Problem

• Extends beer path problem to time-dependent graphs.
• Proposes algorithms for various temporal beer path types.
• Enables dynamic updates like shop openings/closings efficiently.

Innovation

• Introduces beer paths in temporal graphs
• Proposes efficient algorithms for temporal beer paths
• Presents preprocessing for dynamic query answering