Papers by Ralf Borndörfer
Discrete Optimization, 2015
All my work on the subject of this thesis was done during my time at Zuse Institute Berlin (ZIB),... more All my work on the subject of this thesis was done during my time at Zuse Institute Berlin (ZIB), and I would like to thank Martin Grötschel and Ralf Borndörfer for the possibility to work here. I am very grateful to Ralf Borndörfer also for his support, his unlimited number of ideas, and for teaching me a lot about writing mathematics. Furthermore, my time at ZIB would have been very different without my colleagues, especially without those with whom I shared so many enjoyable lunch breaks with interesting conversations. I would like to thank my friend and collaborator Achim Hildenbrandt from the University of Heidelberg for the many inspiring and motivating mathematical discussions we had, and the work we have done together in the last years. Also, my thanks go to Isabel Beckenbach, Achim Hildenbrandt, Linus Mattauch, Sandra de Ruijter, and Edo Schinzinger for critically reading (parts of) this thesis and their very valuable comments. Last but not least, I would like to express my gratitude to my parents for showing me at a very young age how mathematics can be fun. One might consider this the very first step that brought this thesis into existence. The research for this thesis was conducted within the project "Rolling stock roster planning for railways" supported by the DFG Research Center MATHEON. vi xiv cation in Discrete Optimization Journal subject to minor modifications on January 22, 2014 (preprint: [Borndörfer and Heismann, 2012]). A slightly modified version of Section 4.1 has been published as [Heismann and Borndörfer, 2013b]. A slightly modified version of Chapter 5 has been
The line planning problem in public transport deals with the construction of a system of lines th... more The line planning problem in public transport deals with the construction of a system of lines that is both attractive for the passengers and of low costs for the operator. In general, the computed line system should be connected, i.e., for each two stations there have to be a path that is covered by the lines. This subproblem is a generalization of the well-known Steiner tree problem; we call it the Steiner connectivity Problem. We discuss complexity of this problem, generalize the so-called Steiner partition inequalities and give a transformation to the directed Steiner tree problem. We show that directed models provide tight formulations for the Steiner connectivity problem, similar as for the Steiner tree problem.
Modeling, Simulation and Optimization of Complex Processes, 2012
The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem that has ... more The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem that has a natural formulation as an integer quadratic program. This MIQCP is closely related to the Quadratic Assignment Problem and, as far as we know, has not received any attention yet. We show in this article that such a formulation has interesting theoretical properties. Its QP relaxation produces, in particular, the first known nontrivial lower bound on the number of shuntings. In our experiments, it also outperformed the integer linear models computationally. The strengthening technique that raises the lower bound might also be useful for other combinatorial optimization problems.
Discrete Applied Mathematics, 2012
The Vehicle Positioning Problem (VPP), also known as the shunting problem, is a classical combina... more The Vehicle Positioning Problem (VPP), also known as the shunting problem, is a classical combinatorial optimization problem in public transport planning. It has been investigated using several models and approaches, which work well for small instances, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation approach for the VPP and for a multi-period generalization. The main improvement of this model over previous ones is that it provides a tight linear description of the problem that can, in particular, produce non-trivial lower bounds. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial, respectively, pseudo-polynomial time for some versions of the problem. Computational results for largescale instances are reported.
Discrete Applied Mathematics, 2001
This paper investigates a technique of building up discrete relaxations of combinatorial optimiza... more This paper investigates a technique of building up discrete relaxations of combinatorial optimization problems. To establish such a relaxation we introduce a transformation technique-aggregation-that allows one to relax an integer program by means of another integer program. We show that knapsack and set packing relaxations give rise to combinatorial cutting planes in a simple and straightforward way. The constructions are algorithmic.
Annals of Operations Research, 2016
We present the problem of planning mobile tours of inspectors on German motorways to enforce the ... more We present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, we developed a personalized crew rostering model, to schedule the crews of the tours. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The main focus of this paper extends our previous publications on how different requirements for the rostering can be modeled in detail. The second focus is on a bicriterion analysis of the planning problem to find the balance between the control quality and the roster acceptance. Finally, computational results on real-world instances show the practicability of our method and how different input parameters influence the problem complexity.
Mathematical Programming, 2012
The Steiner connectivity problem has the same significance for line planning in public transport ... more The Steiner connectivity problem has the same significance for line planning in public transport as the Steiner tree problem for telecommunication network design. It consists in finding a minimum cost set of elementary paths to connect a subset of nodes in an undirected graph and is, therefore, a generalization of the Steiner tree problem. We propose an extended directed cut formulation for the problem which is, in comparison to the canonical undirected cut formulation, provably strong, implying, e.g., a class of facet defining Steiner partition inequalities. Since a direct application of this formulation is computationally intractable for large instances, we develop a partial projection method to produce a strong relaxation in the space of canonical variables that approximates the extended formulation. We also investigate the separation of Steiner partition inequalities and give computational evidence that these inequalities essentially close the gap between undirected and extended directed cut formulation. Using these techniques, large Steiner connectivity problems with up to 900 nodes can be solved within reasonable optimality gaps of typically less than five percent.
Discrete Applied Mathematics, 2012
The Vehicle Positioning Problem (VPP), also know as the shunting problem, is a classical combinat... more The Vehicle Positioning Problem (VPP), also know as the shunting problem, is a classical combinatorial optimization problem in public transport planning. It has been investigated using several models and approaches, which work well for small instances, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation approach for the VPP and for a multiperiod generalization. The model provides a tight linear description of the problem and can, in particular, produce non-trivial lower bounds. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial resp. pseudo-polynomial time, for some versions of the problem. Computational results for large-scale instances are reported. § The work of this author is supported by CNPq-Brazil. modeled the VPP with two-index variables as a Quadratic Assignment Problem (QAP) and used linearization techniques to solve it as an integer linear program. Hamdouni, Soumis and Desaulniers [9] extend their work exploring robustness and introducing the concept of uniform tracks to solve larger problems. Gallo and Di Miele [7] propose a three-index formulation and extend the problem to deal with vehicles of different lengths and interlaced sequences of arrivals and departures. Freling, Kroon, Lentink, and Huisman [6] and Kroon, Lentink, and Schrijver [12] improved this model by a new formulation of shunting constraints involving additional binary variables. They also consider decomposable vehicles (trains) and different types of tracks (the number of uniform tracks is assumed to be known in advance). Lentink [11] continues by a heuristic and a column generation algorithm based on a decomposition strategy for the problem. Recently, Borndörfer and Cardonha [3] combined the original binary quadratic programming model of Winter with the model improvement of Kroon, Lentink, and Schrijver [12] in order to derive the first non-trivial lower bounds for the number of shunting movements. However, all of the mentioned approaches work only satisfactorily for specifically structured or for very small instances, and, in particular, not for an integrated treatment of multi-period problems.
SIAM Journal on Optimization, 1998
In this paper we investigate whether matrices arising from linear or integer programming problems... more In this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so-called bordered block diagonal form. More precisely, given some matrix A, we try to assign as many rows as possible to some number β of blocks of size κ such that no two rows assigned to different blocks intersect in a common column. Bordered block diagonal form is desirable because it can guide and speed up the solution process for linear and integer programming problems. We show that various matrices from the LP-and MIP-libraries Netlib and Miplib can indeed be decomposed into this form by computing optimal decompositions or decompositions with proven quality. These computations are done with a branch-and-cut algorithm based on polyhedral investigations of the matrix decomposition problem. In practice, however, one would use heuristics to find a good decomposition. We present several heuristic ideas and test their performance. Finally, we investigate the usefulness of optimal matrix decompositions into bordered block diagonal form for integer programming by using such decompositions to guide the branching process in a branch-and-cut code for general mixed integer programs.
The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem that has ... more The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem that has a natural formulation as an integer quadratic program. This MIQCP is closely related to the Quadratic Assignment Problem and, as far as we know, has not received any attention yet. We show in this article that such a formulation has interesting theoretical properties. Its QP relaxation produces, in particular, the first known nontrivial lower bound on the number of shuntings. In our experiments, it also outperformed the integer linear models computationally. The strengthening technique that raises the lower bound might also be useful for other combinatorial optimization problems.
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Papers by Ralf Borndörfer