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

Authors Umang Bhaskar , Katharina Eickhoff , Lennart Kauther , Jannik Matuschke , Britta Peis , Laura Vargas Koch

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Train Routing
  • Scheduling
  • Approximation Algorithms
  • Flows over Time
  • Min-Max Disjoint Paths

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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.

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Umang Bhaskar, Katharina Eickhoff, Lennart Kauther, Jannik Matuschke, Britta Peis, and Laura Vargas Koch. On the Approximability of Train Routing and the Min-Max Disjoint Paths Problem. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.34

Author Details

Umang Bhaskar
  • TIFR Mumbai, India
Katharina Eickhoff
  • RWTH Aachen University, Germany
Lennart Kauther
  • RWTH Aachen University, Germany
Jannik Matuschke
  • KU Leuven, Belgium
Britta Peis
  • RWTH Aachen University, Germany
Laura Vargas Koch
  • RWTH Aachen University, Germany

Funding

  • Bhaskar, Umang: supported by the Department of Atomic Energy, Government of India, under Project No. RTI4001.
  • Eickhoff, Katharina: supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 2236/2.
  • Kauther, Lennart: supported by the German Centre for Rail Traffic Research (DZSF) at the Federal Railway Authority under research project "Methodik der Kapazitätsbewertung des Gesamtsystems und Knotenberechnung".
  • Matuschke, Jannik: supported by Fonds Wetenschappelijk Onderzoek (FWO) under Research Project G072520N "Optimization and Analytics for Stochastic and Robust Project Scheduling".

Acknowledgements

We thank Daniel Schmand, Khai Van Tran and Andreas Wiese for fruitful discussions, as well as Nils Nießen and his group for providing us with foundational insights into train planning. We also thank the participants of the Dagstuhl Seminar 24281 [Correa et al., 2024], the Oberwolfach Workshop in Combinatorial Optimization 2024 [Rothvoss et al., 2024], and the Chile Summer Workshop on Combinatorial Optimization 2025 for many interesting comments and questions. Finally, we thank the three anonymous reviewers for their careful reading of our paper and many useful suggestions.

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