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Documents authored by Schlechte, Thomas

Document
DOI: 10.4230/OASIcs.ATMOS.2022.11
Does Laziness Pay Off? - A Lazy-Constraint Approach to Timetabling

Authors: Torsten Klug, Markus Reuther, and Thomas Schlechte

Published in: OASIcs, Volume 106, 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)

Abstract
Timetabling is a classical and complex task for public transport operators as well as for railway undertakings. The general question is: Which vehicle is taking which route through the transportation network in which order? In this paper, we consider the special setting to find optimal timetables for railway systems under a moving block regime. We directly set up on our work of [T. Schlechte et al., 2022], i.e., we consider the same model formulation and real-world instances of a moving block headway system. In this paper, we present a repair heuristic and a lazy-constraint approach utilizing the callback features of Gurobi, see [Gurobi Optimization, 2022]. We provide an experimental study of the different algorithmic approaches for a railway network with 100 and up to 300 train requests. The computational results show that the lazy-constraint approach together with the repair heuristic significantly improves our previous approaches.
Cite as

Torsten Klug, Markus Reuther, and Thomas Schlechte. Does Laziness Pay Off? - A Lazy-Constraint Approach to Timetabling. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 11:1-11:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{klug_et_al:OASIcs.ATMOS.2022.11,
  author =	{Klug, Torsten and Reuther, Markus and Schlechte, Thomas},
  title =	{{Does Laziness Pay Off? - A Lazy-Constraint Approach to Timetabling}},
  booktitle =	{22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022)},
  pages =	{11:1--11:8},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-259-4},
  ISSN =	{2190-6807},
  year =	{2022},
  volume =	{106},
  editor =	{D'Emidio, Mattia and Lindner, Niels},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2022.11},
  URN =		{urn:nbn:de:0030-drops-171159},
  doi =		{10.4230/OASIcs.ATMOS.2022.11},
  annote =	{Keywords: Moving Block, Railway Track Allocation, Timetabling, Train Routing}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2019.1
A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance

Authors: Boris Grimm, Ralf Borndörfer, Markus Reuther, and Thomas Schlechte

Published in: OASIcs, Volume 75, 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)

Abstract
For providing railway services the company’s railway rolling stock is one if not the most important ingredient. It decides about the number of passenger or cargo trips the company can offer, about the quality a passenger experiences the train ride and it is often related to the image of the company itself. Thus, it is highly desired to have the available rolling stock in the best shape possible. Moreover, in many countries, as Germany where our industrial partner DB Fernverkehr AG (DBF) is located, laws enforce regular vehicle inspections to ensure the safety of the passengers. This leads to rolling stock optimization problems with complex rules for vehicle maintenance. This problem is well studied in the literature for example see [Maróti and Kroon, 2005; Gábor Maróti and Leo G. Kroon, 2007], or [Cordeau et al., 2001] for applications including vehicle maintenance. The contribution of this paper is a new algorithmic approach to solve the Rolling Stock Rotation Problem for the ICE high speed train fleet of DBF with included vehicle maintenance. It is based on a relaxation of a mixed integer linear programming model with an iterative cut generation to enforce the feasibility of a solution of the relaxation in the solution space of the original problem. The resulting mixed integer linear programming model is based on a hypergraph approach presented in [Ralf Borndörfer et al., 2015]. The new approach is tested on real world instances modeling different scenarios for the ICE high speed train network in Germany and compared to the approaches of [Reuther, 2017] that are in operation at DB Fernverkehr AG. The approach shows a significant reduction of the run time to produce solutions with comparable or even better objective function values.
Cite as

Boris Grimm, Ralf Borndörfer, Markus Reuther, and Thomas Schlechte. A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 1:1-1:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{grimm_et_al:OASIcs.ATMOS.2019.1,
  author =	{Grimm, Boris and Bornd\"{o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas},
  title =	{{A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance}},
  booktitle =	{19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)},
  pages =	{1:1--1:12},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-128-3},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{75},
  editor =	{Cacchiani, Valentina and Marchetti-Spaccamela, Alberto},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2019.1},
  URN =		{urn:nbn:de:0030-drops-114136},
  doi =		{10.4230/OASIcs.ATMOS.2019.1},
  annote =	{Keywords: Railway Operations Research, Integer Programming, Infeasible Path Cuts, Cut Separation, Rolling Stock Rotation Problem}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2017.11
Strong Relaxations for the Train Timetabling Problem Using Connected Configurations

Authors: Frank Fischer and Thomas Schlechte

Published in: OASIcs, Volume 59, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)

Abstract
The task of the train timetabling problem or track allocation problem is to find conflict free schedules for a set of trains with predefined routes in a railway network. Especially for non-periodic instances models based on time expanded networks are often used. Unfortunately, the linear programming relaxation of these models is often extremely weak because these models do not describe combinatorial relations like overtaking possibilities very well. In this paper we extend the model by so called connected configuration subproblems. These subproblems perfectly describe feasible schedules of a small subset of trains (2-3) on consecutive track segments. In a Lagrangian relaxation approach we solve several of these subproblems together in order to produce solutions which consist of combinatorially compatible schedules along the track segments. The computational results on a mostly single track corridor taken from the INFORMS RAS Problem Solving Competition 2012 data indicate that our new solution approach is rather strong. Indeed, for this instance the solution of the Lagrangian relaxation is already integral.
Cite as

Frank Fischer and Thomas Schlechte. Strong Relaxations for the Train Timetabling Problem Using Connected Configurations. In 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017). Open Access Series in Informatics (OASIcs), Volume 59, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fischer_et_al:OASIcs.ATMOS.2017.11,
  author =	{Fischer, Frank and Schlechte, Thomas},
  title =	{{Strong Relaxations for the Train Timetabling Problem Using Connected Configurations}},
  booktitle =	{17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)},
  pages =	{11:1--11:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-042-2},
  ISSN =	{2190-6807},
  year =	{2017},
  volume =	{59},
  editor =	{D'Angelo, Gianlorenzo and Dollevoet, Twan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2017.11},
  URN =		{urn:nbn:de:0030-drops-79021},
  doi =		{10.4230/OASIcs.ATMOS.2017.11},
  annote =	{Keywords: combinatorial optimization, train timetabling, Lagrangian relaxation, connected configurations}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2017.15
Cost Projection Methods for the Shortest Path Problem with Crossing Costs

Authors: Marco Blanco, Ralf Borndörfer, Nam Dung Hoàng, Anton Kaier, Pedro M. Casas, Thomas Schlechte, and Swen Schlobach

Published in: OASIcs, Volume 59, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)

Abstract
Real world routing problems, e.g., in the airline industry or in public and rail transit, can feature complex non-linear cost functions. An important case are costs for crossing regions, such as countries or fare zones. We introduce the shortest path problem with crossing costs (SPPCC) to address such situations; it generalizes the classical shortest path problem and variants such as the resource constrained shortest path problem and the minimum label path problem. Motivated by an application in flight trajectory optimization with overflight costs, we focus on the case in which the crossing costs of a region depend only on the nodes used to enter or exit it. We propose an exact Two-Layer-Dijkstra Algorithm as well as a novel cost-projection linearization technique that transforms crossing costs into shadow costs on individual arcs, thus approximating the SPPCC by a standard shortest path problem. We evaluate all algorithms' performance on real-world flight trajectory optimization instances, obtaining very good à posteriori error bounds.
Cite as

Marco Blanco, Ralf Borndörfer, Nam Dung Hoàng, Anton Kaier, Pedro M. Casas, Thomas Schlechte, and Swen Schlobach. Cost Projection Methods for the Shortest Path Problem with Crossing Costs. In 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017). Open Access Series in Informatics (OASIcs), Volume 59, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{blanco_et_al:OASIcs.ATMOS.2017.15,
  author =	{Blanco, Marco and Bornd\"{o}rfer, Ralf and Dung Ho\`{a}ng, Nam and Kaier, Anton and Casas, Pedro M. and Schlechte, Thomas and Schlobach, Swen},
  title =	{{Cost Projection Methods for the Shortest Path Problem with Crossing Costs}},
  booktitle =	{17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017)},
  pages =	{15:1--15:14},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-042-2},
  ISSN =	{2190-6807},
  year =	{2017},
  volume =	{59},
  editor =	{D'Angelo, Gianlorenzo and Dollevoet, Twan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2017.15},
  URN =		{urn:nbn:de:0030-drops-78939},
  doi =		{10.4230/OASIcs.ATMOS.2017.15},
  annote =	{Keywords: shortest path problem, resource constrained shortest path, crossing costs, flight trajectory optimization, overflight fees, cost projection}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2016.12
Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind

Authors: Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Adam Schienle, Thomas Schlechte, and Swen Schlobach

Published in: OASIcs, Volume 54, 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)

Abstract
We study the Flight Planning Problem for a single aircraft, which deals with finding a path of minimal travel time in an airway network. Flight time along arcs is affected by wind speed and direction, which are functions of time. We consider three variants of the problem, which can be modeled as, respectively, a classical shortest path problem in a metric space, a time-dependent shortest path problem with piecewise linear travel time functions, and a time-dependent shortest path problem with piecewise differentiable travel time functions. The shortest path problem and its time-dependent variant have been extensively studied, in particular, for road networks. Airway networks, however, have different characteristics: the average node degree is higher and shortest paths usually have only few arcs. We propose A* algorithms for each of the problem variants. In particular, for the third problem, we introduce an application-specific "super-optimal wind" potential function that overestimates optimal wind conditions on each arc, and establish a linear error bound. We compare the performance of our methods with the standard Dijkstra algorithm and the Contraction Hierarchies (CHs) algorithm. Our computational results on real world instances show that CHs do not perform as well as on road networks. On the other hand, A* guided by our potentials yields very good results. In particular, for the case of piecewise linear travel time functions, we achieve query times about 15 times shorter than CHs.
Cite as

Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Adam Schienle, Thomas Schlechte, and Swen Schlobach. Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind. In 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016). Open Access Series in Informatics (OASIcs), Volume 54, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blanco_et_al:OASIcs.ATMOS.2016.12,
  author =	{Blanco, Marco and Bornd\"{o}rfer, Ralf and Hoang, Nam-Dung and Kaier, Anton and Schienle, Adam and Schlechte, Thomas and Schlobach, Swen},
  title =	{{Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind}},
  booktitle =	{16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016)},
  pages =	{12:1--12:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-021-7},
  ISSN =	{2190-6807},
  year =	{2016},
  volume =	{54},
  editor =	{Goerigk, Marc and Werneck, Renato F.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2016.12},
  URN =		{urn:nbn:de:0030-drops-65360},
  doi =		{10.4230/OASIcs.ATMOS.2016.12},
  annote =	{Keywords: shortest path problem, A*, flight trajectory optimization, preprocessing, contraction hierarchies, time-dependent shortest path problem}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2014.79
A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation Problem

Authors: Ralf Borndörfer, Markus Reuther, and Thomas Schlechte

Published in: OASIcs, Volume 42, 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2014)

Abstract
We propose a new coarse-to-fine approach to solve certain linear programs by column generation. The problems that we address contain layers corresponding to different levels of detail, i.e., coarse layers as well as fine layers. These layers are utilized to design efficient pricing rules. In a nutshell, the method shifts the pricing of a fine linear program to a coarse counterpart. In this way, major decisions are taken in the coarse layer, while minor details are tackled within the fine layer. We elucidate our methodology by an application to a complex railway rolling stock rotation problem. We provide comprehensive computational results that demonstrate the benefit of this new technique for the solution of large scale problems.
Cite as

Ralf Borndörfer, Markus Reuther, and Thomas Schlechte. A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation Problem. In 14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 42, pp. 79-91, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{borndorfer_et_al:OASIcs.ATMOS.2014.79,
  author =	{Bornd\"{o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas},
  title =	{{A Coarse-To-Fine Approach to the Railway Rolling Stock Rotation  Problem}},
  booktitle =	{14th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{79--91},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-75-0},
  ISSN =	{2190-6807},
  year =	{2014},
  volume =	{42},
  editor =	{Funke, Stefan and Mihal\'{a}k, Mat\'{u}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2014.79},
  URN =		{urn:nbn:de:0030-drops-47549},
  doi =		{10.4230/OASIcs.ATMOS.2014.79},
  annote =	{Keywords: Coarse-To-Fine Linear Programming, Rolling Stock Rotation Problem}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2011.146
A Hypergraph Model for Railway Vehicle Rotation Planning

Authors: Ralf Borndörfer, Markus Reuther, Thomas Schlechte, and Steffen Weider

Published in: OASIcs, Volume 20, 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2011)

Abstract
We propose a model for the integrated optimization of vehicle rotations and vehicle compositions in long distance railway passenger transport. The main contribution of the paper is a hypergraph model that is able to handle the challenging technical requirements as well as very general stipulations with respect to the "regularity" of a schedule. The hypergraph model directly generalizes network flow models, replacing arcs with hyperarcs. Although NP-hard in general, the model is computationally well-behaved in practice. High quality solutions can be produced in reasonable time using high performance Integer Programming techniques, in particular, column generation and rapid branching. We show that, in this way, large-scale real world instances of our cooperation partner DB Fernverkehr can be solved.
Cite as

Ralf Borndörfer, Markus Reuther, Thomas Schlechte, and Steffen Weider. A Hypergraph Model for Railway Vehicle Rotation Planning. In 11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. Open Access Series in Informatics (OASIcs), Volume 20, pp. 146-155, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{borndorfer_et_al:OASIcs.ATMOS.2011.146,
  author =	{Bornd\"{o}rfer, Ralf and Reuther, Markus and Schlechte, Thomas and Weider, Steffen},
  title =	{{A Hypergraph Model for Railway Vehicle Rotation Planning}},
  booktitle =	{11th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems},
  pages =	{146--155},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-33-0},
  ISSN =	{2190-6807},
  year =	{2011},
  volume =	{20},
  editor =	{Caprara, Alberto and Kontogiannis, Spyros},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2011.146},
  URN =		{urn:nbn:de:0030-drops-32746},
  doi =		{10.4230/OASIcs.ATMOS.2011.146},
  annote =	{Keywords: Rolling Stock Planning, Hypergraph Modeling, Integer Programming, Column Generation, Rapid Branching}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2010.13
Railway Track Allocation by Rapid Branching

Authors: Ralf Borndörfer, Thomas Schlechte, and Steffen Weider

Published in: OASIcs, Volume 14, 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10) (2010)

Abstract
The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.
Cite as

Ralf Borndörfer, Thomas Schlechte, and Steffen Weider. Railway Track Allocation by Rapid Branching. In 10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10). Open Access Series in Informatics (OASIcs), Volume 14, pp. 13-23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{borndorfer_et_al:OASIcs.ATMOS.2010.13,
  author =	{Bornd\"{o}rfer, Ralf and Schlechte, Thomas and Weider, Steffen},
  title =	{{Railway Track Allocation by Rapid Branching}},
  booktitle =	{10th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS'10)},
  pages =	{13--23},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-20-0},
  ISSN =	{2190-6807},
  year =	{2010},
  volume =	{14},
  editor =	{Erlebach, Thomas and L\"{u}bbecke, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2010.13},
  URN =		{urn:nbn:de:0030-drops-27465},
  doi =		{10.4230/OASIcs.ATMOS.2010.13},
  annote =	{Keywords: track allocation problem, integer programming, rapid branching heuristic, proximal bundle method}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2007.1170
05. Models for Railway Track Allocation

Authors: Ralf Borndörfer and Thomas Schlechte

Published in: OASIcs, Volume 7, 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07) (2007)

Abstract
The optimal track allocation problem (OPTRA) is to find, in a given railway network, a conflict free set of train routes of maximum value. We study two types of integer programming formulations for this problem: a standard formulation that models block conflicts in terms of packing constraints, and a novel formulation of the `extended' type that is based on additional `configuration' variables. The packing constraints in the standard formulation stem from an interval graph and can therefore be separated in polynomial time. It follows that the LP-relaxation of a strong version of this model, including all clique inequalities from block conflicts, can be solved in polynomial time. We prove that the LP-relaxation of the extended formulation can also be solved in polynomial time, and that it produces the same LP-bound. Albeit the two formulations are in this sense equivalent, the extended formulation has advantages from a computational point of view. It features a constant number of rows and is amenable to standard column generation techniques. Results of an empirical model comparison on mesoscopic data for the Hanover-Fulda-Kassel region of the German long distance railway network are reported.
Cite as

Ralf Borndörfer and Thomas Schlechte. 05. Models for Railway Track Allocation. In 7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07). Open Access Series in Informatics (OASIcs), Volume 7, pp. 62-78, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{borndorfer_et_al:OASIcs.ATMOS.2007.1170,
  author =	{Bornd\"{o}rfer, Ralf and Schlechte, Thomas},
  title =	{{05. Models for Railway Track Allocation}},
  booktitle =	{7th Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems (ATMOS'07)},
  pages =	{62--78},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-04-0},
  ISSN =	{2190-6807},
  year =	{2007},
  volume =	{7},
  editor =	{Ahuja, Ravindra K. and Liebchen, Christian and Mesa, Juan A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2007.1170},
  URN =		{urn:nbn:de:0030-drops-11701},
  doi =		{10.4230/OASIcs.ATMOS.2007.1170},
  annote =	{Keywords: Track allocation, train timetabling,integer programming, column generation}
}
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