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DOI: 10.4230/OASIcs.ATMOS.2025.4

Visualization of Event Graphs for Train Schedules

Authors: Johann Hartleb, Marie Schmidt, Samuel Wolf, and Alexander Wolff

Published in: OASIcs, Volume 137, 25th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2025)

Abstract

Train timetables can be represented as event graphs, where events correspond to a train passing through a location at a certain point in time. A visual representation of an event graph is important for many applications such as dispatching and (the development of) dispatching software. A common way to represent event graphs are time-space diagrams. In such a diagram, key locations are visualized on the y-axis and time on the x-axis of a coordinate system. A train’s movement is then represented as a connected sequence of line segments in this coordinate system. This visualization allows for an easy detection of infrastructure conflicts and safety distance violations. However, time-space diagrams are usually used only to depict event graphs that are restricted to corridors, where an obvious ordering of the locations exists. In this paper, we consider the visualization of general event graphs in time-space diagrams, where the challenge is to find an ordering of the locations that produces readable drawings. We argue that this means to minimize the number of turns, i.e., the total number of changes in y-direction. To this end, we establish a connection between this problem and Maximum Betweenness. Then we develop a preprocessing strategy to reduce the instance size. We also propose a parameterized algorithm and integer linear programming formulations. We experimentally evaluate the preprocessing strategy and the integer programming formulations on a real-world dataset. Our best algorithm solves every instance in the dataset in less than a second. This suggests that turn-optimal time-space diagrams can be computed in real time.

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Johann Hartleb, Marie Schmidt, Samuel Wolf, and Alexander Wolff. Visualization of Event Graphs for Train Schedules. In 25th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2025). Open Access Series in Informatics (OASIcs), Volume 137, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hartleb_et_al:OASIcs.ATMOS.2025.4,
  author =	{Hartleb, Johann and Schmidt, Marie and Wolf, Samuel and Wolff, Alexander},
  title =	{{Visualization of Event Graphs for Train Schedules}},
  booktitle =	{25th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2025)},
  pages =	{4:1--4:20},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-404-8},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{137},
  editor =	{Sauer, Jonas and Schmidt, Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2025.4},
  URN =		{urn:nbn:de:0030-drops-247607},
  doi =		{10.4230/OASIcs.ATMOS.2025.4},
  annote =	{Keywords: Graph Drawing, Event Graphs, Integer Linear Programming, Parameterized Algorithms, Treewidth}
}

Document

DOI: 10.4230/OASIcs.ATMOS.2020.15

A Rolling Horizon Heuristic with Optimality Guarantee for an On-Demand Vehicle Scheduling Problem

Authors: Johann Hartleb and Marie Schmidt

Published in: OASIcs, Volume 85, 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)

Abstract

We consider a basic vehicle scheduling problem that arises in the context of travel demand models: Given demanded vehicle trips, what is the minimal number of vehicles needed to fulfill the demand? In this paper, we model the vehicle scheduling problem as a network flow problem. Since instances arising in the context of travel demand models are often so big that the network flow model becomes intractable, we propose using a rolling horizon heuristic to split huge problem instances into smaller subproblems and solve them independently to optimality. By letting the horizons of the subproblems overlap, it is possible to look ahead to the demand of the next subproblem. We prove that composing the solutions of the subproblems yields an optimal solution to the whole problem if the overlap of the horizons is sufficiently large. Our experiments show that this approach is not only suitable for solving extremely large instances that are intractable as a whole, but it is also possible to decrease the solution time for large instances compared to a comprehensive approach.

Cite as

Johann Hartleb and Marie Schmidt. A Rolling Horizon Heuristic with Optimality Guarantee for an On-Demand Vehicle Scheduling Problem. In 20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020). Open Access Series in Informatics (OASIcs), Volume 85, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hartleb_et_al:OASIcs.ATMOS.2020.15,
  author =	{Hartleb, Johann and Schmidt, Marie},
  title =	{{A Rolling Horizon Heuristic with Optimality Guarantee for an On-Demand Vehicle Scheduling Problem}},
  booktitle =	{20th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2020)},
  pages =	{15:1--15:18},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-170-2},
  ISSN =	{2190-6807},
  year =	{2020},
  volume =	{85},
  editor =	{Huisman, Dennis and Zaroliagis, Christos D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2020.15},
  URN =		{urn:nbn:de:0030-drops-131513},
  doi =		{10.4230/OASIcs.ATMOS.2020.15},
  annote =	{Keywords: Rolling Horizon Heuristic, Vehicle Scheduling, Network Flow}
}

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Visualization of Event Graphs for Train Schedules

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A Rolling Horizon Heuristic with Optimality Guarantee for an On-Demand Vehicle Scheduling Problem

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