TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Klostermeier, Fridolin A1 - Küttner, Christian T1 - Telebus Berlin - Mobilität für Behinderte N2 - Steigendes Mobilitätsbedürfnis der Behinderten bei angespannter Haushaltslage --- diese Situation ergab vor drei Jahren beim Berliner Behindertenfahrdienst {\em Telebus\/} Handlungsbedarf. Gleichzeitig stie\ss{} die verwendete manuelle (Fahrzeug-)Tourenplanung mit damals etwa 1.000 Fahrtwünschen pro Tag an ihre Grenzen. Zur Lösung dieser Probleme war der effiziente Einsatz von Computern und von {\em mathematischen Optimierungsverfahren\/} erforderlich. Ergebnis des daraufhin gestarteten {\em Telebus-Projektes\/} ist die Entwicklung eines neuen, alle Arbeitsabläufe unterstützenden Computersystems, das seit über einem Jahr im Einsatz ist. Herzstück dieses Systems ist ein auf mathematischen Methoden der ganzzahligen Optimierung basierendes Verfahren zur Tourenplanung. Ziel dieses Artikels ist die Beschreibung dieses Verfahrens und seiner Verwendung bei der Behindertenbeförderung. Das Beispiel Telebus zeigt, da\ss{} der Einsatz mathematischer Optimierungstechniken neue Möglichkeiten für Kosteneinsparungen bei gleichzeitiger Serviceverbesserung auch in anderen Bereichen des ÖPNV eröffnet. T3 - ZIB-Report - SC-96-40 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2500 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grimm, Boris A1 - Reuther, Markus A1 - Schlechte, Thomas T1 - Optimization of Handouts for Rolling Stock Rotations Visualization N2 - A railway operator creates (rolling stock) rotations in order to have a precise master plan for the operation of a timetable by railway vehicles. A rotation is considered as a cycle that multiply traverses a set of operational days while covering trips of the timetable. As it is well known, the proper creation of rolling stock rotations by, e.g., optimization algorithms is challenging and still a topical research subject. Nevertheless, we study a completely different but strongly related question in this paper, i.e.: How to visualize a rotation? For this purpose, we introduce a basic handout concept, which directly leads to the visualization, i.e., handout of a rotation. In our industrial application at DB Fernverkehr AG, the handout is exactly as important as the rotation itself. Moreover, it turns out that also other European railway operators use exactly the same methodology (but not terminology). Since a rotation can have many handouts of different quality, we show how to compute optimal ones through an integer program (IP) by standard software. In addition, a construction as well as an improvement heuristic are presented. Our computational results show that the heuristics are a very reliable standalone approach to quickly find near-optimal and even optimal handouts. The efficiency of the heuristics is shown via a computational comparison to the IP approach. T3 - ZIB-Report - ZR-16-73 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-61430 SN - 1438-0064 ER - TY - JOUR A1 - Harrod, Steven A1 - Schlechte, Thomas T1 - A Direct Comparison of Physical Block Occupancy Versus Timed Block Occupancy in Train Timetabling Formulations JF - Transportation Research Part E: Logistics and Transportation Review N2 - Two fundamental mathematical formulations for railway timetabling are compared on a common set of sample problems, representing both multiple track high density services in Europe and single track bidirectional operations in North America. One formulation, ACP, enforces against conflicts by constraining time intervals between trains, while the other formulation, RCHF, monitors physical occupation of controlled track segments. The results demonstrate that both ACP and RCHF return comparable solutions in the aggregate, with some significant differences in select instances, and a pattern of significant differences in performance and constraint enforcement overall. Y1 - 2013 U6 - https://doi.org/10.1016/j.tre.2013.04.003 VL - 54 SP - 50 EP - 66 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Karbstein, Marika A1 - Pfetsch, Marc T1 - Models for Fare Planning in Public Transport JF - Discrete Applied Mathematics Y1 - 2012 U6 - https://doi.org/10.1016/j.dam.2012.02.027 VL - 160 IS - 18 SP - 2591 EP - 2605 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Schlechte, Thomas T1 - Re-optimizing ICE Rotations after a Tunnel Breakdown near Rastatt N2 - Planning rolling stock movements in industrial passenger railway applications isa long-term process based on timetables which are also often valid for long periods of time. For these timetables and rotation plans, i.e., plans of railway vehicle movements are constructed as templates for these periods. During operation the rotation plans are affected by all kinds of unplanned events. An unusal example for that is the collapse of a tunnel ceiling near Rastatt in southern Germany due to construction works related to the renewal of the central station in Stuttgart. As a result the main railway connection between Stuttgart and Frankfurt am Main, located on top of the tunnel, had to be closed from August 12th to October 2nd 2017. This had a major impact on the railway network in southern Germany. Hence, all rotation plans and train schedules for both passenger and cargo traffic had to be revised. In this paper we focus on a case study for this situation and compute new rotation plans via mixed integer programming for the ICE high speed fleet of DB Fernverkehr AG one of the largest passenger railway companies in Europe. In our approach we take care of some side constraints to ensure a smooth continuation of the rotation plans after the disruption has ended. T3 - ZIB-Report - 19-02 KW - case study KW - re-optimization KW - railway rolling stock optimization KW - mixed integer programming Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-73976 SN - 1438-0064 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Grimm, Boris A1 - Schlechte, Thomas T1 - Re-optimizing ICE Rotations after a Tunnel Breakdown near Rastatt T2 - Proceedings of the 8th International Conference on Railway Operations Modelling and Analysis - RailNorrköping 2019 N2 - Planning rolling stock movements in industrial passenger railway applications isa long-term process based on timetables which are also often valid for long periods of time. For these timetables and rotation plans, i.e., plans of railway vehicle movements are constructed as templates for these periods. During operation the rotation plans are affected by all kinds of unplanned events. An unusal example for that is the collapse of a tunnel ceiling near Rastatt in southern Germany due to construction works related to the renewal of the central station in Stuttgart. As a result the main railway connection between Stuttgart and Frankfurt am Main, located on top of the tunnel, had to be closed from August 12th to October 2nd 2017. This had a major impact on the railway network in southern Germany. Hence, all rotation plans and train schedules for both passenger and cargo traffic had to be revised. In this paper we focus on a case study for this situation and compute new rotation plans via mixed integer programming for the ICE high speed fleet of DB Fernverkehr AG one of the largest passenger railway companies in Europe. In our approach we take care of some side constraints to ensure a smooth continuation of the rotation plans after the disruption has ended. Y1 - 2019 UR - http://www.ep.liu.se/ecp/article.asp?issue=069&article=011&volume= SN - 978-91-7929-992-7 SN - 1650-3686 VL - Linköping Electronic Conference Proceedings IS - 069 SP - 160 EP - 168 PB - Linköping University Electronic Press, Linköpings universitet CY - Linköping, Sweden ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Grimm, Boris A1 - Reuther, Markus A1 - Schlechte, Thomas T1 - Optimization of handouts for rolling stock rotations JF - Journal of Rail Transport Planning & Management N2 - A railway operator creates (rolling stock) rotations in order to have a precise master plan for the operation of a timetable by railway vehicles. A rotation is considered as a cycle that multiply traverses a set of operational days while covering trips of the timetable. As it is well known, the proper creation of rolling stock rotations by, e.g., optimization algorithms is challenging and still a topical research subject. Nevertheless, we study a completely different but strongly related question in this paper, i.e.: How to visualize a rotation? For this purpose, we introduce a basic handout concept, which directly leads to the visualization, i.e., handout of a rotation. In our industrial application at DB Fernverkehr AG, the handout is exactly as important as the rotation itself. Moreover, it turns out that also other European railway operators use exactly the same methodology (but not terminology). Since a rotation can have many handouts of different quality, we show how to compute optimal ones through an integer program (IP) by standard software. In addition, a construction as well as an improvement heuristic are presented. Our computational results show that the heuristics are a very reliable standalone approach to quickly find near-optimal and even optimal handouts. The efficiency of the heuristics is shown via a computational comparison to the IP approach. Y1 - 2019 U6 - https://doi.org/10.1016/j.jrtpm.2019.02.001 IS - 10 SP - 1 EP - 8 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Schulz, Christof A1 - Weider, Steffen T1 - The Rolling Stock Rotation Planning Problem under Revenue Considerations N2 - In many railway undertakings a railway timetable is offered that is valid for a longer period of time. At DB Fernverkehr AG, one of our industrial partners, this results in a summer and a winter timetable. For both of these timetables rotation plans, i.e., a detailed plan of railway vehicle movements is constructed as a template for this period. Sometimes there are be periods where you know for sure that vehicle capacities are not sufficient to cover all trips of the timetable or to transport all passenger of the trips. Reasons for that could be a heavy increase of passenger flow, a heavy decrease of vehicle availability, impacts from nature, or even strikes of some employees. In such events the rolling stock rotations have to be adapted. Optimization methods are particularly valuable in such situations in order to maintain a best possible level of service or to maximize the expected revenue using the resources that are still available. In most cases found in the literature, a rescheduling based on a timetable update is done, followed by the construction of new rotations that reward the recovery of parts of the obsolete rotations. We consider a different, novel, and more integrated approach. The idea is to guide the cancellation of the trips or reconfiguration of the vehicle composition used to operate a trip of the timetable by the rotation planning process, which is based on the mixed integer programming approach presented in Reuther (2017). The goal is to minimize the operating costs while cancelling or operating a trip with an insufficient vehicle configuration in sense of passenger capacities inflicts opportunity costs and loss of revenue, which are based on an estimation of the expected number of passengers. The performance of the algorithms presented in two case studies, including real world scenarios from DB Fernverkehr AG and a railway operator in North America. T3 - ZIB-Report - 19-01 Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-71339 SN - 1438-0064 ER - TY - CHAP A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Schulz, Christof A1 - Weider, Steffen T1 - The Rolling Stock Rotation Planning Problem under Revenue Considerations T2 - Proceedings of the Rail Transport Demand Management Conference N2 - In many railway undertakings a railway timetable is offered that is valid for a longer period of time. At DB Fernverkehr AG, one of our industrial partners, this results in a summer and a winter timetable. For both of these timetables rotation plans, i.e., a detailed plan of railway vehicle movements is constructed as a template for this period. Sometimes there are be periods where you know for sure that vehicle capacities are not sufficient to cover all trips of the timetable or to transport all passenger of the trips. Reasons for that could be a heavy increase of passenger flow, a heavy decrease of vehicle availability, impacts from nature, or even strikes of some employees. In such events the rolling stock rotations have to be adapted. Optimization methods are particularly valuable in such situations in order to maintain a best possible level of service or to maximize the expected revenue using the resources that are still available. In most cases found in the literature, a rescheduling based on a timetable update is done, followed by the construction of new rotations that reward the recovery of parts of the obsolete rotations. We consider a different, novel, and more integrated approach. The idea is to guide the cancellation of the trips or reconfiguration of the vehicle composition used to operate a trip of the timetable by the rotation planning process, which is based on the mixed integer programming approach presented in Reuther (2017). The goal is to minimize the operating costs while cancelling or operating a trip with an insufficient vehicle configuration in sense of passenger capacities inflicts opportunity costs and loss of revenue, which are based on an estimation of the expected number of passengers. The performance of the algorithms presented in two case studies, including real world scenarios from DB Fernverkehr AG and a railway operator in North America. Y1 - 2018 ER - TY - GEN A1 - Beckenbach, Isabel A1 - Hatzel, Meike A1 - Wiederrecht, Sebastian T1 - The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs N2 - The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on the tight cut decomposition is that the list of bricks obtained is the same independent of the choice of tight cuts made during the tight cut decomposition procedure. This implies that finding a tight cut decomposition is polynomial time equivalent to finding a single tight cut. We generalise the notions of a tight cut, a tight cut contraction and a tight cut decomposition to hypergraphs. By providing an example, we show that the outcome of the tight cut decomposition on general hypergraphs is no longer unique. However, we are able to prove that the uniqueness of the tight cut decomposition is preserved on a slight generalisation of uniform hypergraphs. Moreover, we show how the tight cut decomposition leads to a decomposition of the perfect matching polytope of uniformable hypergraphs and that the recognition problem for tight cuts in uniformable hypergraphs is polynomial time solvable. T3 - ZIB-Report - 18-61 KW - uniform hypergraphs KW - perfect matchings KW - tight cuts Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-71149 SN - 1438-0064 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Olthoff, Mats T1 - A Solution Approach to the Vehicle Routing Problem with Perishable Goods N2 - This paper focuses on a special case of vehicle routing problem where perishable goods are considered. Deliveries have to be performed until a due date date, which may vary for different products. Storing products is prohibited. Since late deliveries have a direct impact on the revenues for these products, a precise demand prediction is important. In our practical case the product demands and vehicle driving times for the product delivery are dependent on weather conditions, i.e., temperatures, wind, and precipitation. In this paper the definition and a solution approach to the Vehicle Routing Problem with Perishable Goods is presented. The approach includes a procedure how historical weather data is used to predict demands and driving times. Its run time and solution quality is evaluated on different data sets given by the MOPTA Competition 2018. T3 - ZIB-Report - 20-03 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-77020 SN - 1438-0064 ER - TY - CHAP A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Olthoff, Mats T1 - A Solution Approach to the Vehicle Routing Problem with Perishable Goods T2 - Operations Research 2019 Proceedings N2 - This paper focuses on a special case of vehicle routing problem where perishable goods are considered. Deliveries have to be performed until a due date date, which may vary for different products. Storing products is prohibited. Since late deliveries have a direct impact on the revenues for these products, a precise demand prediction is important. In our practical case the product demands and vehicle driving times for the product delivery are dependent on weather conditions, i.e., temperatures, wind, and precipitation. In this paper the definition and a solution approach to the Vehicle Routing Problem with Perishable Goods is presented. The approach includes a procedure how historical weather data is used to predict demands and driving times. Its run time and solution quality is evaluated on different data sets given by the MOPTA Competition 2018. Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-77020 ER - TY - GEN A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - The Cone of Flow Matrices: Approximation Hierarchies and Applications N2 - 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. T3 - ZIB-Report - 17-32 KW - Flows in graphs KW - Approximation Hierarchies KW - Copositive Programming KW - Semidefinite Programming Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64399 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - A Primal-Dual Approximation Algorithm for the Steiner Connectivity Problem N2 - 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. T3 - ZIB-Report - 13-54 KW - Primal-Dual Approximation KW - Steiner Connectivity Problem KW - Degree Property KW - Hypergraph Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-42430 SN - 1438-0064 ER - TY - THES A1 - Löbel, Fabian T1 - Solving Integrated Timetabling and Passenger Routing Problems Using the Modulo Network Simplex Algorithm N2 - 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. Y1 - 2017 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas T1 - A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance T2 - 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019) N2 - 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 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. T3 - ZIB-Report - 19-61 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-75501 SN - 1438-0064 ER - TY - CHAP A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas ED - Cacchiani, Valentina ED - Marchetti-Spaccamela, Alberto T1 - A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance T2 - 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019) N2 - 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. Y1 - 2019 UR - https://drops.dagstuhl.de/opus/volltexte/2019/11413/ U6 - https://doi.org/10.4230/OASIcs.ATMOS.2019.1 VL - 75 SP - 1:1 EP - 1:12 PB - Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik CY - Dagstuhl, Germany ER - TY - GEN A1 - Tesch, Alexander T1 - A Nearly Exact Propagation Algorithm for Energetic Reasoning in O(n^2 log n) N2 - 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. T3 - ZIB-Report - 16-25 KW - Cumulative Scheduling KW - Energetic Reasoning KW - Sweep Line Algorithm Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59332 SN - 1438-0064 ER - TY - JOUR A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - The Cone of Flow Matrices: Approximation Hierarchies and Applications JF - Networks N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1002/net.21820 VL - 72 IS - 1 SP - 128 EP - 150 ER - TY - JOUR A1 - Beckenbach, Isabel A1 - Borndörfer, Ralf T1 - Hall’s and Kőnig’s theorem in graphs and hypergraphs JF - Discrete Mathematics N2 - 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–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs. Y1 - 2018 U6 - https://doi.org/10.1016/j.disc.2018.06.013 VL - 341 IS - 10 SP - 2753 EP - 2761 ER - TY - GEN A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - Approximation Hierarchies for the cone of flow matrices N2 - 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. T3 - ZIB-Report - 18-20 KW - Flows in graphs KW - Approximation hierarchies KW - Copositive programming Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68424 SN - 1438-0064 ER - TY - CHAP A1 - Sagnol, Guillaume A1 - Blanco, Marco A1 - Sauvage, Thibaut T1 - Approximation Hierarchies for the cone of flow matrices T2 - INOC 2017 – 8th International Network Optimization Conference N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1016/j.endm.2018.02.002 VL - 64 SP - 275 EP - 284 ER - TY - CHAP A1 - Schade, Stanley A1 - Schlechte, Thomas A1 - Witzig, Jakob T1 - Structure-based Decomposition for Pattern-Detection for Railway Timetables T2 - Operations Research Proceedings 2017 N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1007/978-3-319-89920-6_95 SP - 715 EP - 721 PB - Springer International Publishing ER - TY - GEN A1 - Borndörfer, Ralf A1 - Euler, Ricardo A1 - Karbstein, Marika A1 - Mett, Fabian T1 - Ein mathematisches Modell zur Beschreibung von Preissystemen im öV N2 - Dieses Dokument fasst den Stand der mathematischen Modellierung von Preissystemen des öV mittels eines am ZIB entwickelten Tarifgraphenmodells zusammen. Damit sind sehr einfache und konzise Beschreibungen von Tarifstrukturen möglich, die sich algorithmisch behandeln lassen: Durch das zeitgleiche Tracken eines Pfades im Routinggraphen im Tarifgraphen kann schon während einer Routenberechnung der Preis bestimmt werden. Wir beschreiben zunächst das Konzept. Die konkrete Realisierung wird im Folgenden beispielhaft an den Tarifsystemen der Verkehrsverbünde Warnow, MDV, Vogtland, Bremen/Niedersachsen, Berlin/Brandenburg und Mittelsachsen erläutert. Anschließend folgen Überlegungen zur konkreten Implementierung von Kurzstrecken-Tarifen und zur Behandlung des Verbundübergriffs. T3 - ZIB-Report - 18-47 Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-70564 SN - 1438-0064 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Breuer, Matthias A1 - Grimm, Boris A1 - Reuther, Markus A1 - Schade, Stanley A1 - Schlechte, Thomas T1 - Timetable Sparsification by Rolling Stock Rotation Optimization T2 - Operations Research 2017 N2 - Rolling stock optimization is a task that naturally arises by operating a railway system. It could be seen with different level of details. From a strategic perspective to have a rough plan which types of fleets to be bought to a more operational perspective to decide which coaches have to be maintained first. This paper presents a new approach to deal with rolling stock optimisation in case of a (long term) strike. Instead of constructing a completely new timetable for the strike period, we propose a mixed integer programming model that is able to choose appropriate trips from a given timetable to construct efficient tours of railway vehicles covering an optimized subset of trips, in terms of deadhead kilometers and importance of the trips. The decision which trip is preferred over the other is made by a simple evaluation method that is deduced from the network and trip defining data. Y1 - 2018 U6 - https://doi.org/10.1007/978-3-319-89920-6_96 SP - 723 EP - 728 PB - Springer International Publishing ER - TY - CHAP A1 - Beckenbach, Isabel ED - Kliewer, Natalia ED - Ehmke, Jan Fabian ED - Borndörfer, Ralf T1 - A Hypergraph Network Simplex Algorithm T2 - Operations Research Proceedings 2017 N2 - We describe a network simplex algorithm for the minimum cost flow problem on graph-based hypergraphs which are directed hypergraphs of a particular form occurring in railway rotation planning. The algorithm is based on work of Cambini, Gallo, and Scutellà who developed a hypergraphic generalization of the network simplex algorithm. Their main theoretical result is the characterization of basis matrices. We give a similar characterization for graph-based hypergraphs and show that some operations of the simplex algorithm can be done combinatorially by exploiting the underlying digraph structure. Y1 - 2018 VL - Operations Research Proceedings 2017 SP - 309 EP - 316 PB - Springer CY - Cham ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Pfetsch, Marc T1 - Models for Line Planning in Public Transport N2 - The \emph{line planning problem} is one of the fundamental problems in strategic planning of public and rail transport. It consists of finding lines and corresponding frequencies in a public transport network such that a given travel demand can be satisfied. There are (at least) two objectives. The transport company wishes to minimize its operating cost; the passengers request short travel times. We propose two new multi-commodity flow models for line planning. Their main features, in comparison to existing models, are that the passenger paths can be freely routed and that the lines are generated dynamically. T3 - ZIB-Report - 04-10 KW - line planning KW - column generation Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7854 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc ED - Barnhart, Cynthia ED - Clausen, Uwe ED - Lauther, Ulrich ED - Möhring, Rolf T1 - Line Planning and Connectivity T2 - Models and Algorithms for Optimization in Logistics Y1 - 2009 IS - 09261 PB - Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany CY - Dagstuhl, Germany ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Klostermeier, Fridolin A1 - Küttner, Christian T1 - Berliner Telebussystem bietet Mobilität für Behinderte JF - Der Nahverkehr Y1 - 1997 VL - 1/2 SP - 20 EP - 22 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Neumann, Marika A1 - Pfetsch, Marc ED - Haasis, Hans-Dietrich ED - Kopfer, Herbert ED - Schönberger, Jörn T1 - Optimal Fares for Public Transport T2 - Operations Research Proceedings 2005 Y1 - 2006 UR - {http://opus.kobv.de/zib/volltexte/2005/869/} SP - 29 EP - 36 PB - Springer-Verlag ER - TY - CHAP A1 - Neumann, Marika ED - Waldmann, Karl-Heinz ED - Stocker, Ulrike T1 - Fare Planning for Public Transport T2 - Operations Research Proceedings 2006 Y1 - 2007 UR - {http://opus.kobv.de/zib/volltexte/2009/1160/} SP - 61 EP - 66 PB - Springer-Verlag ER - TY - JOUR A1 - Neumann, Marika T1 - Mathematische Preisplanung im ÖPNV JF - OR News Y1 - 2007 IS - 30 SP - 29 EP - 31 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Klostermeier, Fridolin A1 - Küttner, Christian T1 - Optimierung des Berliner Behindertenfahrdienstes JF - DMV-Mitteilungen Y1 - 1997 VL - 2 SP - 38 EP - 43 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Schlechte, Thomas A1 - Weider, Steffen T1 - Railway Track Allocation by Rapid Branching N2 - 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. T3 - ZIB-Report - 10-22 KW - Trassenallokationsproblem KW - Ganzzahlige Programmierung KW - Rapid-Branching-Heuristik KW - Proximale Bündelmethode KW - Optimization KW - Railway Track Allocation KW - Bundle Method KW - Rapid Branching Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11864 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Karbstein, Marika T1 - Metric Inequalities for Routings on Direct Connections N2 - We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We show that the concept of metric inequalities to characterize capacities that support a multi-commodity flow can be generalized to deal with direct connections. T3 - ZIB-Report - 14-04 KW - metric inequalities KW - direct connection KW - multi-commodity flow problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-44219 SN - 1438-0064 ER - TY - CHAP A1 - Heismann, Olga A1 - Borndörfer, Ralf T1 - The Random Hypergraph Assignment Problem T2 - Proceedings of the 16th International Multiconference INFORMATION SOCIETY - IS 2013 N2 - The analysis of random instances of a combinatorial optimization problem, especially their optimal values, can provide a better insight into its structure. Such an extensive analysis was theoretically and practically done for the assignment problem ("random assignment problem") and several of its generalizations. For a recent generalization of the assignment problem to bipartite hypergraphs, the hypergraph assignment problem, such results do not exist so far. We consider a random version of the hypergraph assignment problem for the simplest possible complete bipartite hypergraphs. They have only edges and proper hyperedges of size four and follow a special structure, but the hypergraph assignment problem for this type of hypergraphs is, however, already NP-hard. It can be viewed as a combination of two assignment problems. For random hyperedge costs exponentially i.i.d. with mean 1 we show computational results that suggest that the expected value of minimum cost hyperassignments converges to some value around 1.05 with a small standard deviation. The computational results also suggest that the optimal value is most probably attained with half of the maximum possible number of proper hyperedges. The main result of this paper is the proof that the expected value of a minimum cost hyperassignment which uses exactly half the possible maximum number of proper hyperedges if the vertex number tends to infinity lies between 0.3718 and 1.8310 when hyperedge costs are exponentially i.i.d. with mean 1. Y1 - 2013 SP - 599 EP - 602 ER - TY - CHAP A1 - Sagnol, Guillaume T1 - Network-related problems in optimal experimental design and second order cone programming T2 - Proceedings of PROBASTAT'2011, Tatra Mountains Mathematical Publications N2 - In the past few years several applications of optimal experimental designs have emerged to optimize the measurements in communication networks. The optimal design problems arising from this kind of applications share three interesting properties: (i) measurements are only available at a small number of locations of the network; (ii) each monitor can simultaneously measure several quantities, which can be modeled by ``multiresponse experiments"; (iii) the observation matrices depend on the topology of the network. In this paper, we give an overview of these experimental design problems and recall recent results for the computation of optimal designs by Second Order Cone Programming (SOCP). New results for the network-monitoring of a discrete time process are presented. In particular, we show that the optimal design problem for the monitoring of an AR1 process can be reduced to the standard form and we give experimental results. Y1 - 2012 U6 - https://doi.org/10.2478/v10127-012-0016-x VL - 51 SP - 161 EP - 171 ER - TY - JOUR A1 - Sagnol, Guillaume T1 - On the semidefinite representation of real functions applied to symmetric matrices JF - Linear Algebra and its Applications N2 - We present a new semidefinite representation for the trace of a real function f applied to symmetric matrices, when a semidefinite representation of the convex function f is known. Our construction is intuitive, and yields a representation that is more compact than the previously known one. We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices that for a rational number p in the interval (0,1], the matrix X raised to the exponent p is the largest element of a set represented by linear matrix inequalities. We give numerical results for a problem inspired from the theory of experimental designs, which show that the new semidefinite programming formulation yields a speed-up factor in the order of 10. Y1 - 2013 U6 - https://doi.org/10.1016/j.laa.2013.08.021 VL - 439 IS - 10 SP - 2829 EP - 2843 ER - TY - CHAP A1 - Hildenbrandt, Achim A1 - Reinelt, Gerhard A1 - Heismann, Olga T1 - Integer Programming Models for the Target Visitation Problem T2 - Proceedings of the 16th International Multiconference INFORMATION SOCIETY - IS 2013 N2 - The target visitation problem (TVP) is concerned with finding a route to visit a set of targets starting from and returning to some base. In addition to the distance traveled a tour is evaluated by taking also preferences into account which address the sequence in which the targets are visited. The problem thus is a combination of two well-known combinato- rial optimization problems: the traveling salesman and the linear ordering problem. In this paper we present several possible IP-Models for this problem and compared them to their usability for branch-and-cut approaches. Y1 - 2013 SP - 569 EP - 572 ER - TY - CHAP A1 - Heismann, Olga A1 - Borndörfer, Ralf T1 - Minimum Cost Hyperassignments with Applications to ICE/IC Rotation Planning T2 - Operations Research Proceedings 2011 N2 - Vehicle rotation planning is a fundamental problem in rail transport. It decides how the railcars, locomotives, and carriages are operated in order to implement the trips of the timetable. One important planning requirement is operational regularity, i.e., using the rolling stock in the same way on every day of operation. We propose to take regularity into account by modeling the vehicle rotation planning problem as a minimum cost hyperassignment problem (HAP). Hyperassignments are generalizations of assignments from directed graphs to directed hypergraphs. Finding a minimum cost hyperassignment is NP-hard. Most instances arising from regular vehicle rotation planning, however, can be solved well in practice. We show that, in particular, clique inequalities strengthen the canonical LP relaxation substantially. Y1 - 2012 SP - 59 EP - 64 ER - TY - JOUR A1 - Langenhan, Andreas A1 - Borndörfer, Ralf A1 - Löbel, Andreas A1 - Schulz, Christof A1 - Weider, Steffen ED - Muñoz, J. C. ED - Voß, S. T1 - Duty Scheduling Templates JF - Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12) N2 - We propose duty templates as a novel concept to produce similar duty schedules for similar days of operation in public transit. Duty templates can conveniently handle various types of similarity requirements, and they can be implemented with ease using standard algorithmic techniques. They have produced good results in practice. Y1 - 2012 ER - TY - THES A1 - Karbstein, Marika T1 - Line Planning and Connectivity N2 - This thesis introduces the Steiner connectivity problem. It is a generalization of the well known Steiner tree problem. Given a graph G = (V, E) and a subset T ⊆ V of the nodes, the Steiner tree problem consists in finding a cost minimal set of edges connecting all nodes in T . The Steiner connectivity problem chooses, instead of edges, from a given set of paths a subset to connect all nodes in T . We show in the first part of this thesis that main results about complexity, approximation, integer programming formulations, and polyhedra can be generalized from the Steiner tree problem to the Steiner connectivity problem. An example for a straightforward generalization are the Steiner partition inequalities, a fundamental class of facet defining inequalities for the Steiner tree problem. They can be defined for the Steiner connectivity problem in an analogous way as for the Steiner tree problem. An example for a generalization that needs more effort is the definition of a directed cut formulation and the proof that this dominates the canonical undirected cut formulation enriched by all Steiner partition inequalities. For the Steiner connectivity problem this directed cut formulation leads to extended formulations, a concept that is not necessary for the Steiner tree problem. There are also major differences between both problems. For instance, the case T = V for the Steiner connectivity problem is equivalent to a set covering problem and, hence, not a polynomial solvable case as in the Steiner tree problem. The Steiner connectivity problem is not only an interesting generalization of the Steiner tree problem but also the underlying connectivity problem in line planning with inte- grated passenger routing. The integrated line planning and passenger routing problem is an important planning problem in service design of public transport and the topic of the second part. Given is the infrastructure network of a public transport system where the edges correspond to streets and tracks and the nodes correspond to stations/stops of lines. The task is to find paths in the infrastructure network for lines and passengers such that the capacities of the lines suffice to transport all passengers. Existing models in the literature that integrate a passenger routing in line planning either treat transfers in a rudimentary way and, hence, neglect an important aspect for the choice of the pas- senger routes, or they treat transfers in a too comprehensive way and cannot be solved for large scale real world problems. We propose a new model that focuses on direct connections. The attractiveness of transfer free connections is increased by introducing a transfer penalty for each non-direct connection. In this way, a passenger routing is computed that favors direct connections. For the computation of this model we also implemented algorithms influenced by the results for the Steiner connectivity problem. We can compute with our model good solutions that minimize a weighted sum of line operating costs and passengers travel times. These solutions improve the solutions of an existing approach, that does not consider direct connections, by up to 17%. In contrast to a comprehensive approach, that considers every transfer and for which we could not even solve the root LP within 10 hours for large instances, the solutions of the new model, computed in the same time, are close to optimality (<1%) or even optimal for real world instances. In a project with the Verkehr in Potsdam GmbH to compute the line plan for 2010 we showed that our approach is applicable in practice and can be used to solve real world problems. Y1 - 2013 UR - www.zib.de/karbstein/scplpp.pdf SN - 978-3-8439-1062-0 ER - TY - JOUR A1 - Hiller, Benjamin A1 - Klug, Torsten A1 - Tuchscherer, Andreas T1 - An Exact Reoptimization Algorithm for the Scheduling of Elevator Groups JF - Flexible Services and Manufacturing Journal Y1 - 2013 U6 - https://doi.org/10.1007/s10696-013-9175-6 SP - 1 EP - 24 ER - TY - CHAP A1 - Schlechte, Thomas T1 - Railway Track Allocation BT - Selected Papers of the International Annual Conference of the German Operations Research Society (GOR), Leibniz University of Hannover, Germany, September 5-7, 2012 T2 - Operations Research Proceedings Y1 - 2012 U6 - https://doi.org/10.1007/978-3-319-00795-314 SP - 15 EP - 20 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Karbstein, Marika A1 - Pfetsch, Marc T1 - The Steiner connectivity problem JF - Mathematical Programming A Y1 - 2013 U6 - https://doi.org/10.1007/s10107-012-0564-5 VL - 142 IS - 1 SP - 133 EP - 167 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Grötschel, Martin A1 - Klostermeier, Fridolin A1 - Küttner, Christian T1 - Telebus-Disposition: Ein Konzept zur Serviceverbesserung bei gleichzeitiger Kostensenkung.[nur hausintern] N2 - "`Telebus"' ist der soziale Behindertenfahrdienst im Land Berlin. Das Telebus-Forschungsprojekt des Konrad-Zuse-Zentrums für Informationstechnik (ZIB) hat das Ziel, den Fahrdienst (insbesondere die Disposition der Telebusse) zu verbessern, d.h. kundenfreundlicher zu gestalten und gleichzeitig billiger zu machen. In diesem Bericht werden die bisherigen Ergebnisse dargestellt und weitere Möglichkeiten zur Verbesserung des Service und zur Reduzierung der Kosten skizziert. \originalTeX T3 - ZIB-Report - TR-93-03 Y1 - 1993 N1 - Volltext nur Hausintern ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Weider, Steffen ED - Muñoz, J. C. ED - Voß, S. T1 - Vehicle Rotation Planning for Intercity Railways JF - Proceedings of Conference on Advanced Systems for Public Transport 2012 (CASPT12) N2 - This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers. Y1 - 2012 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Löbel, Andreas A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Weider, Steffen ED - Muñoz, J. C. ED - Voß, S. T1 - Rapid Branching JF - Public Transport N2 - We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning. Y1 - 2013 VL - 5 IS - 1 SP - 3 EP - 23 PB - Springer Berlin Heidelberg ER - TY - CHAP A1 - Scheumann, René A1 - Fügenschuh, Armin A1 - Schenker, Sebastian A1 - Vierhaus, Ingmar A1 - Borndörfer, Ralf A1 - Finkbeiner, Matthias ED - Seliger, Günther T1 - Global Manufacturing: How to Use Mathematical Optimisation Methods to Transform to Sustainable Value Creation BT - (GCSM 2012) T2 - Proceedings of the 10th Global Conference on Sustainable Manufacturing Y1 - 2012 SN - 978-605-63463-1-6 SP - 538 EP - 545 ER - TY - JOUR A1 - Sagnol, Guillaume T1 - Approximation of a maximum-submodular-coverage problem involving spectral functions, with application to experimental designs JF - Discrete Applied Mathematics N2 - We study a family of combinatorial optimization problems defined by a parameter $p\in[0,1]$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as $p$ goes to $0$, and to a trivial instance of the knapsack problem as $p$ goes to $1$. In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to $1$ as $p$ goes to $1$. Y1 - 2013 U6 - https://doi.org/10.1016/j.dam.2012.07.016 VL - 161 IS - 1-2 SP - 258 EP - 276 ER - TY - JOUR A1 - Sagnol, Guillaume T1 - A class of Semidefinite Programs with rank-one solutions JF - Linear Algebra and its Applications N2 - We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP. The optimization problems of this class are semidefinite packing problems, which are the SDP analogs to vector packing problems. Of particular interest is the case in which our result guarantees the existence of a solution of rank one: we show that the computation of this solution actually reduces to a Second Order Cone Program (SOCP). We point out an application in statistics, in the optimal design of experiments. Y1 - 2011 U6 - https://doi.org/10.1016/j.laa.2011.03.027 VL - 435 IS - 6 SP - 1446 EP - 1463 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Schlechte, Thomas A1 - Swarat, Elmar T1 - Railway Track Allocation -- Simulation, Aggregation, and Optimization JF - Proc. 1st International Workshop on High-speed and Intercity Railways (IWHIR 2011) N2 - Today the railway timetabling process and the track allocation is one of the most challenging problems to solve by a railway company. Especially due to the deregulation of the transport market in the recent years several suppliers of railway traffic have entered the market in Europe. This leads to more potential conflicts between trains caused by an increasing demand of train paths. Planning and operating railway transportation systems is extremely hard due to the combinatorial complexity of the underlying discrete optimization problems, the technical intricacies, and the immense size of the problem instances. In order to make best use of the infrastructure and to ensure economic operation, efficient planning of the railway operation is indispensable. Mathematical optimization models and algorithms can help to automatize and tackle these challenges. Our contribution in this paper is to present a renewed planning process due to the liberalization in Europe and an associated concept for track allocation, that consists of three important parts, simulation, aggregation, and optimization. Furthermore, we present results of our general framework for real world data. Y1 - 2012 U6 - https://doi.org/10.1007/978-3-642-27963-8 VL - 2 IS - 148 SP - 53 EP - 70 ER -