@misc{SagnolBlancoSauvage2017, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {The Cone of Flow Matrices: Approximation Hierarchies and Applications}, issn = {1438-0064}, doi = {10.1002/net.21820}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64399}, year = {2017}, abstract = {Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices \$\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}\$, where \$\vec{1}_P\in\RR^n\$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over \$\mathcal{K}\$. This cone is intractable: we prove that the membership problem associated to \$\mathcal{K}\$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix \$X\in \operatorname{span} \mathcal{K}\$ as a linear combination of some \$\vec{1}_P\vec{1}_P^T\$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.}, language = {en} } @misc{BorndoerferKarbstein2013, author = {Bornd{\"o}rfer, Ralf and Karbstein, Marika}, title = {A Primal-Dual Approximation Algorithm for the Steiner Connectivity Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42430}, year = {2013}, abstract = {We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree property for terminal nodes in hypergraphs which generalizes the well-known graph property that the average degree of terminal nodes in Steiner trees is at most 2.}, language = {en} } @masterthesis{Loebel2017, type = {Bachelor Thesis}, author = {L{\"o}bel, Fabian}, title = {Solving Integrated Timetabling and Passenger Routing Problems Using the Modulo Network Simplex Algorithm}, year = {2017}, abstract = {Common models and solving approaches for the problem of periodic timetabling, that is, determining periodic arrival and departure times of a given public transportation network's lines so that the total weighted travel time of all passengers is minimized, fail to take passenger behavior into account. Current research is attempting to resolve this issue and first results show a positive impact on solution quality. This thesis aims to give a brief overview of literature on the topic of timetabling and to then state a heuristic approach to Integrated Timetabling by blending passenger behavior into the Modulo Network Simplex algorithm, which is part of the research carried out by the author's study group at the Zuse Institut Berlin in the framework of MATHEON's research project MI-3 supported by the Einstein Foundation Berlin.}, language = {en} } @misc{GrimmBorndoerferReutheretal.2020, 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}, journal = {19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)}, issn = {1438-0064}, doi = {https://doi.org/10.4230/OASIcs.ATMOS.2019.1}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-75501}, year = {2020}, 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 Maroti and Kroon 2005, 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 Bornd{\"o}rfer 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.}, language = {en} } @inproceedings{GrimmBorndoerferReutheretal.2019, 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}, volume = {75}, booktitle = {19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)}, editor = {Cacchiani, Valentina and Marchetti-Spaccamela, Alberto}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}, address = {Dagstuhl, Germany}, doi = {10.4230/OASIcs.ATMOS.2019.1}, pages = {1:1 -- 1:12}, year = {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 [Mar{\´o}ti and Kroon, 2005; G{\´a}bor Mar{\´o}ti 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 Bornd{\"o}rfer 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.}, language = {en} } @misc{Tesch2016, author = {Tesch, Alexander}, title = {A Nearly Exact Propagation Algorithm for Energetic Reasoning in O(n^2 log n)}, issn = {1438-0064}, doi = {10.1007/978-3-319-44953-1_32}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59332}, year = {2016}, abstract = {In constraint programming, energetic reasoning constitutes a powerful start time propagation rule for cumulative scheduling problems (CuSP). In this paper, we first present an improved time interval checking algorithm that is derived from a polyhedral model. In a second step, we extend this algorithm to an energetic reasoning propagation algorithm with complexity O(n^2 log n) where n denotes the number of jobs. The key idea is based on a new sweep line subroutine that efficiently evaluates the relevant time intervals for all jobs. In particular, our algorithm yields at least one possible energetic reasoning propagation for each job. Finally, we show that on the vast number of relevant time intervals our approach yields the maximum possible propagation according to the energetic reasoning rule.}, language = {en} } @article{SagnolBlancoSauvage2018, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {The Cone of Flow Matrices: Approximation Hierarchies and Applications}, volume = {72}, journal = {Networks}, number = {1}, doi = {10.1002/net.21820}, pages = {128 -- 150}, year = {2018}, abstract = {Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone generated by the matrices \$\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}\$, where \$\vec{1}_P\in\RR^n\$ is the incidence vector of the (s,t)-path P. We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation of a flow, reduce to a linear optimization problem over \$\mathcal{K}\$. This cone is intractable: we prove that the membership problem associated to \$\mathcal{K}\$ is NP-complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial-time the decomposition of a matrix \$X\in \operatorname{span} \mathcal{K}\$ as a linear combination of some \$\vec{1}_P\vec{1}_P^T\$'s. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of~K. We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc-capacities.}, language = {en} } @article{BeckenbachBorndoerfer2018, author = {Beckenbach, Isabel and Bornd{\"o}rfer, Ralf}, title = {Hall's and Kőnig's theorem in graphs and hypergraphs}, volume = {341}, journal = {Discrete Mathematics}, number = {10}, doi = {10.1016/j.disc.2018.06.013}, pages = {2753 -- 2761}, year = {2018}, abstract = {We investigate the relation between Hall's theorem and Kőnig's theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall's theorem, thereby showing that this class strictly contains all Kőnig-Egerv{\´a}ry graphs. Furthermore, we give a generalization of Hall's theorem to normal hypergraphs.}, language = {en} } @misc{SagnolBlancoSauvage2018, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {Approximation Hierarchies for the cone of flow matrices}, issn = {1438-0064}, doi = {10.1016/j.endm.2018.02.002}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68424}, year = {2018}, abstract = {Let \$G\$ be a directed acyclic graph with \$n\$ arcs, a source \$s\$ and a sink \$t\$. We introduce the cone \$K\$ of flow matrices, which is a polyhedral cone generated by the matrices \$1_P 1_P^T \in R^{n\times n}\$, where \$1_P\in R^n\$ is the incidence vector of the \$(s,t)\$-path \$P\$. Several combinatorial problems reduce to a linear optimization problem over \$K\$. This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a completely positive representation of \$K\$. We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.}, language = {en} } @inproceedings{SagnolBlancoSauvage2018, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {Approximation Hierarchies for the cone of flow matrices}, volume = {64}, booktitle = {INOC 2017 - 8th International Network Optimization Conference}, doi = {10.1016/j.endm.2018.02.002}, pages = {275 -- 284}, year = {2018}, abstract = {Let \$G\$ be a directed acyclic graph with \$n\$ arcs, a source \$s\$ and a sink \$t\$. We introduce the cone \$K\$ of flow matrices, which is a polyhedral cone generated by the matrices \$1_P 1_P^T \in R^{n\times n}\$, where \$1_P\in R^n\$ is the incidence vector of the \$(s,t)\$-path \$P\$. Several combinatorial problems reduce to a linear optimization problem over \$K\$. This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a completely positive representation of \$K\$. We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.}, language = {en} } @inproceedings{SchadeSchlechteWitzig2018, author = {Schade, Stanley and Schlechte, Thomas and Witzig, Jakob}, title = {Structure-based Decomposition for Pattern-Detection for Railway Timetables}, booktitle = {Operations Research Proceedings 2017}, publisher = {Springer International Publishing}, doi = {10.1007/978-3-319-89920-6_95}, pages = {715 -- 721}, year = {2018}, abstract = {We consider the problem of pattern detection in large scale railway timetables. This problem arises in rolling stock optimization planning in order to identify invariant sections of the timetable for which a cyclic rotation plan is adequate. We propose a dual reduction technique which leads to an decomposition and enumeration method. Computational results for real world instances demonstrate that the method is able to produce optimal solutions as fast as standard MIP solvers.}, language = {en} } @misc{BorndoerferEulerKarbsteinetal.2018, author = {Bornd{\"o}rfer, Ralf and Euler, Ricardo and Karbstein, Marika and Mett, Fabian}, title = {Ein mathematisches Modell zur Beschreibung von Preissystemen im {\"o}V}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-70564}, year = {2018}, abstract = {Dieses Dokument fasst den Stand der mathematischen Modellierung von Preissystemen des {\"o}V mittels eines am ZIB entwickelten Tarifgraphenmodells zusammen. Damit sind sehr einfache und konzise Beschreibungen von Tarifstrukturen m{\"o}glich, die sich algorithmisch behandeln lassen: Durch das zeitgleiche Tracken eines Pfades im Routinggraphen im Tarifgraphen kann schon w{\"a}hrend einer Routenberechnung der Preis bestimmt werden. Wir beschreiben zun{\"a}chst das Konzept. Die konkrete Realisierung wird im Folgenden beispielhaft an den Tarifsystemen der Verkehrsverb{\"u}nde Warnow, MDV, Vogtland, Bremen/Niedersachsen, Berlin/Brandenburg und Mittelsachsen erl{\"a}utert. Anschließend folgen {\"U}berlegungen zur konkreten Implementierung von Kurzstrecken-Tarifen und zur Behandlung des Verbund{\"u}bergriffs.}, language = {de} }