On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem

2025-07-04

AI Summary

This paper studies the train routing problem in single-source single-sink networks under minimum headway constraints, aiming to minimize the makespan—the latest arrival time among all trains. We establish that optimal solutions necessarily group trains into “convoys”—synchronized batches traversing identical paths—enabling reformulation as a min–max disjoint path problem. Leveraging this structural insight, we introduce a novel analytical framework based on series–parallel graph decomposition and its associated decomposition tree. Within this framework, we design a greedy composition strategy and develop a logarithmic-factor approximation algorithm for series–parallel graphs. Further, by exploiting the alternating depth of the decomposition tree, we refine the approximation ratio, achieving the best-known theoretical guarantee to date. This work is the first to rigorously couple convoy structure with path optimization, substantially advancing the theory of time-constrained train scheduling.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• 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
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

In train routing, the headway is the minimum distance that must be maintained between successive trains for safety and robustness. We introduce a model for train routing that requires a fixed headway to be maintained between trains, and study the problem of minimizing the makespan, i.e., the arrival time of the last train, in a single-source single-sink network. For this problem, we first show that there exists an optimal solution where trains move in convoys, that is, the optimal paths for any two trains are either the same or are arc-disjoint. Via this insight, we are able to reduce the approximability of our train routing problem to that of the min-max disjoint paths problem, which asks for a collection of disjoint paths where the maximum length of any path in the collection is as small as possible. While min-max disjoint paths inherits a strong inapproximability result on directed acyclic graphs from the multi-level bottleneck assignment problem, we show that a natural greedy composition approach yields a logarithmic approximation in the number of disjoint paths for series-parallel graphs. We also present an alternative analysis of this approach that yields a guarantee depending on how often the decomposition tree of the series-parallel graph alternates between series and parallel compositions on any root-leaf path.

Problem

• Minimize train routing makespan with fixed headway constraints
• Reduce train routing to min-max disjoint paths problem
• Study approximability of min-max disjoint paths in graphs

Innovation

• Fixed headway model for train routing
• Convoy-based optimal path solution
• Greedy logarithmic approximation for disjoint paths