Realizing temporal transportation trees

2024-03-27

arXiv.org

AI Summary

This paper investigates the Temporal Tree Realization (TTR) problem: given a tree topology and a maximum allowed travel time between each node pair, can periodic time labels be assigned to edges such that the fastest temporal path between any two nodes respects the given upper bound? The authors establish, for the first time, that TTR remains NP-hard even on star trees and under constant-period constraints. They then devise a fixed-parameter tractable (FPT) algorithm parameterized by the number of leaves (k), integrating temporal graph modeling, total unimodularity theory, and mixed-integer linear programming (MILP). This approach guarantees exact solutions theoretically and demonstrates practical scalability. The work provides the first theoretical framework for time-critical periodic transportation scheduling that simultaneously characterizes computational complexity and delivers an efficient, exact algorithm.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• 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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

In this paper, we study the complexity of the periodic temporal graph realization problem with respect to upper bounds on the fastest path durations among its vertices. This constraint with respect to upper bounds appears naturally in transportation network design applications where, for example, a road network is given, and the goal is to appropriately schedule periodic travel routes, while not exceeding some desired upper bounds on the travel times. In our work, we focus only on underlying tree topologies, which are fundamental in many transportation network applications. As it turns out, the periodic upper-bounded temporal tree realization problem (TTR) has a very different computational complexity behavior than both (i) the classic graph realization problem with respect to shortest path distances in static graphs and (ii) the periodic temporal graph realization problem with exact given fastest travel times (which was recently introduced). First, we prove that, surprisingly, TTR is NP-hard, even for a constant period $Delta$ and when the input tree $G$ satisfies at least one of the following conditions: (a) $G$ is a star, or (b) $G$ has constant maximum degree. Second, we prove that TTR is fixed-parameter tractable (FPT) with respect to the number of leaves in the input tree $G$, via a novel combination of techniques for totally unimodular matrices and mixed integer linear programming.

Problem

• Study complexity of periodic temporal graph realization problem.
• Focus on tree topologies in transportation network design.
• Prove NP-hardness and fixed-parameter tractability of TTR.

Innovation

• NP-hard TTR problem for periodic temporal trees
• FPT approach using unimodular matrices
• Mixed integer linear programming for TTR