TY - JOUR A1 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika A1 - Lindner, Niels T1 - Separation of cycle inequalities in periodic timetabling JF - Discrete Optimization N2 - Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem in public transport. We give the first pseudo-polynomial time separation algorithm for cycle inequalities, and we contribute a rigorous proof for the pseudo-polynomial time separability of the change-cycle inequalities. Moreover, we provide several NP-completeness results, indicating that pseudo-polynomial time is best possible. The efficiency of these cutting planes is demonstrated on real-world instances of the periodic timetabling problem. Y1 - 2020 U6 - https://doi.org/10.1016/j.disopt.2019.100552 IS - 35 SP - 100552 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Lindner, Niels A1 - Roth, Sarah T1 - A Concurrent Approach to the Periodic Event Scheduling Problem JF - Journal of Rail Transport Planning & Management N2 - We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib. Y1 - 2020 U6 - https://doi.org/10.1016/j.jrtpm.2019.100175 SN - 2210-9706 IS - 15 SP - 100175 ER - TY - JOUR A1 - Prause, Felix A1 - Hoppmann-Baum, Kai A1 - Defourny, Boris A1 - Koch, Thorsten T1 - The maximum diversity assortment selection problem JF - Mathematical Methods of Operations Research N2 - In this article, we introduce the Maximum Diversity Assortment Selection Problem (MDASP), which is a generalization of the two-dimensional Knapsack Problem (2D-KP). Given a set of rectangles and a rectangular container, the goal of 2D-KP is to determine a subset of rectangles that can be placed in the container without overlapping, i.e., a feasible assortment, such that a maximum area is covered. MDASP is to determine a set of feasible assortments, each of them covering a certain minimum threshold of the container, such that the diversity among them is maximized. Thereby, diversity is defined as the minimum or average normalized Hamming distance of all assortment pairs. MDASP was the topic of the 11th AIMMS-MOPTA Competition in 2019. The methods described in this article and the resulting computational results won the contest. In the following, we give a definition of the problem, introduce a mathematical model and solution approaches, determine upper bounds on the diversity, and conclude with computational experiments conducted on test instances derived from the 2D-KP literature. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1007/s00186-021-00740-2 VL - 93 SP - 521 EP - 554 PB - Mathematical Methods of Operations Research ER - TY - JOUR A1 - Riedmüller, Stephanie A1 - Lindner, Niels T1 - Column Generation for Periodic Timetabling N2 - Periodic timetabling for public transportation networks is typically modelled as a Periodic Event Scheduling Problem (PESP). Solving instances of the benchmark library PESPlib to optimality continues to pose a challenge. As a further approach towards this goal, we remodel the problem by a time discretization of the underlying graph and consider arc-based as well as path-based integer programming formulations. For the path-based case, we use cycles on the graph expansion of the operational lines as variables and, therefore, include more of the problem inherent structure into the model. A consequence is the validity of several known inequalities and a lower bound on the LP-relaxation, that is the best known to date. As an extension we integrate passenger routing into the new model. The proposed models have an advantage in the linear programming relaxation, on the one hand, but have an increased problem size, on the other hand. We define the corresponding pricing problems for the use of column generation to handle the size. Both models are practically tested on different problem instances. Y1 - 2025 ER - TY - JOUR A1 - van Lieshout, Rolf A1 - Lindner, Niels T1 - A Compact Cycle Formulation for the Multiperiodic Event Scheduling Problem N2 - The Periodic Event Scheduling Problem (PESP) is a fundamental model in periodic timetabling for public transport systems, assuming a common period across all events. However, real-world networks often feature heterogeneous service frequencies. This paper studies the Multiperiodic Event Scheduling Problem (MPESP), a generalization of PESP that allows each event to recur at its own individual period. While more expressive, MPESP presents new modeling challenges due to the loss of a global period. We present a cycle-based formulation for MPESP that extends the strongest known formulation for PESP and, in contrast to existing approaches, is valid for any MPESP instance. Crucially, the formulation requires a cycle basis derived from a spanning tree satisfying specific structural properties, which we formalize and algorithmically construct, extending the concept of sharp spanning trees to rooted instances. We further prove a multiperiodic analogue of the cycle periodicity property. Our new formulation solves nearly all tested instances, including several large-scale real-world public transport networks, to optimality or with small optimality gaps, dramatically outperforming existing arc-based models. The results demonstrate the practical potential of MPESP in capturing heterogeneous frequencies without resorting to artificial event duplication. Y1 - 2025 ER - TY - JOUR A1 - Maristany de las Casas, Pedro A1 - Sedeño-Noda, Antonio A1 - Borndörfer, Ralf T1 - New Dynamic Programming Algorithm for the Multiobjective Minimum Spanning Tree Problem JF - Computers & Operations Research N2 - The Multiobjective Minimum Spanning Tree (MO-MST) problem is a variant of the Minimum Spanning Tree problem, in which the costs associated with every edge of the input graph are vectors. In this paper, we design a new dynamic programming MO-MST algorithm. Dynamic programming for a MO-MST instance leads to the definition of an instance of the One-to-One Multiobjective Shortest Path (MOSP) problem and both instances have equivalent solution sets. The arising MOSP instance is defined on a so called transition graph. We study the original size of this graph in detail and reduce its size using cost dependent arc pruning criteria. To solve the MOSP instance on the reduced transition graph, we design the Implicit Graph Multiobjective Dijkstra Algorithm (IG-MDA), exploiting recent improvements on MOSP algorithms from the literature. All in all, the new IG-MDA outperforms the current state of the art on a big set of instances from the literature. Our code and results are publicly available. Y1 - 2025 U6 - https://doi.org/10.1016/j.cor.2024.106852 VL - 173 SP - 106852 ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Bortoletto, Enrico T1 - Computing All Shortest Passenger Routes with a Tropical Dijkstra Algorithm JF - EURO Journal on Transportation and Logistics N2 - Given a public transportation network, which and how many passenger routes can potentially be shortest paths, when all possible timetables are taken into account? This question leads to shortest path problems on graphs with interval costs on their arcs and is closely linked to multi-objective optimization. We introduce a Dijkstra algorithm based on polynomials over the tropical semiring that computes complete or minimal sets of efficient paths. We demonstrate that this approach is computationally feasible by employing it on the public transport network of the city of Wuppertal and instances of the benchmarking set TimPassLib, and we evaluate the resulting sets of passenger routes. Y1 - 2025 U6 - https://doi.org/10.1016/j.ejtl.2025.100163 VL - 14 ER - TY - JOUR A1 - Grimm, Boris A1 - Hoogervorst, Rowan A1 - Borndörfer, Ralf T1 - A Comparison of Two Models for Rolling Stock Scheduling JF - Transportation Science N2 - A major step in the planning process of passenger railway operators is the assignment of rolling stock, that is, train units, to the trips of the timetable. A wide variety of mathematical optimization models have been proposed to support this task, which we discuss and argue to be justified in order to deal with operational differences between railway operators, and hence different planning requirements, in the best possible way. Our investigation focuses on two commonly used models, the composition model and the hypergraph model, that were developed for Netherlands Railways (NS) and DB Fernverkehr AG (DB), respectively. We compare these models in two distinct problem settings, an NS setting and DB-light setting and consider different model variants to tune the models to these settings. We prove that in both of these settings, the linear programming bounds of the two models are equally strong as long as a number of reasonable assumptions are met. However, through a numerical evaluation on NS and DB-light instances, we show that the numerical performance of the models strongly depends on the instances. Although the composition model is the most compact and fastest model for the NS instances, an adjusted version of this model grows quickly for the DB-light instances and is then outperformed by the considered hypergraph model variants. Moreover, we show that a depot-extended version of the hypergraph model is able to combine strengths of both models and show good performance on both the NS and DB-light instances. Y1 - 2025 U6 - https://doi.org/10.1287/trsc.2024.0505 VL - 59 IS - 5 ER - TY - JOUR A1 - Maristany de las Casas, Pedro A1 - Sedeño-Noda, Antonio A1 - Borndörfer, Ralf A1 - Huneshagen, Max T1 - K-Shortest Simple Paths Using Biobjective Path Search JF - Mathematical Programming Computation N2 - In this paper we introduce a new algorithm for the k-Shortest Simple Paths (K-SSP) problem with an asymptotic running time matching the state of the art from the literature. It is based on a black-box algorithm due to Roditty and Zwick (2012) that solves at most 2k instances of the Second Shortest Simple Path (2-SSP) problem without specifying how this is done. We fill this gap using a novel approach: we turn the scalar 2-SSP into instances of the Biobjective Shortest Path problem. Our experiments on grid graphs and on road networks show that the new algorithm is very efficient in practice. Y1 - 2025 U6 - https://doi.org/10.1007/s12532-025-00276-0 VL - 17 SP - 349 EP - 384 ER - TY - JOUR A1 - Lindner, Niels A1 - Masing, Berenike T1 - On the split closure of the periodic timetabling polytope JF - Mathematical Programming N2 - The Periodic Event Scheduling Problem (PESP) is the central mathematical tool for periodic timetable optimization in public transport. PESP can be formulated in several ways as a mixed-integer linear program with typically general integer variables. We investigate the split closure of these formulations and show that split inequalities are identical with the recently introduced flip inequalities. While split inequalities are a general mixed-integer programming technique, flip inequalities are defined in purely combinatorial terms, namely cycles and arc sets of the digraph underlying the PESP instance. It is known that flip inequalities can be separated in pseudo-polynomial time. We prove that this is best possible unless P = NP, but also observe that the complexity becomes linear-time if the cycle defining the flip inequality is fixed. Moreover, introducing mixed-integer-compatible maps, we compare the split closures of different formulations, and show that reformulation or binarization by subdivision do not lead to stronger split closures. Finally, we estimate computationally how much of the optimality gap of the instances of the benchmark library PESPlib can be closed exclusively by split cuts, and provide better dual bounds for five instances. KW - Periodic event scheduling problem KW - Periodic timetabling KW - Split closure KW - Mixed-integer programming Y1 - 2025 U6 - https://doi.org/10.1007/s10107-025-02220-5 SN - 0025-5610 PB - Springer Science and Business Media LLC ER - TY - JOUR A1 - Euler, Ricardo A1 - Maristany de las Casas, Pedro T1 - Labeling methods for partially ordered paths JF - European Journal of Operational Research N2 - The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions’ properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems’ weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs. Y1 - 2024 U6 - https://doi.org/10.1016/j.ejor.2024.05.002 VL - 318 IS - 1 SP - 19 EP - 30 PB - Elsevier BV ET - 1 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Euler, Ricardo A1 - Karbstein, Marika T1 - Ein Graphen-basiertes Modell zur Beschreibung von Preissystemen im öffentlichen Nahverkehr JF - HEUREKA 21 N2 - In dieser Arbeit wird ein graphenbasiertes Modell zur Einbindung von Preissystemen des öffentlichen Nahverkehrs in Routing-Algorithmen vorgestellt. Jeder Knoten des Graphen repräsentiert einen abstrakten Preiszustand einer Route und ist an einen tatsächlichen Preis gekoppelt. Damit sind sehr einfache und konzise Beschreibungen von Tarifstrukturen möglich, diesich algorithmisch behandeln lassen. Durch das zeitgleiche Tracken eines Pfades im Routinggraphen im Ticketgraphen kann schon während einer Routenberechnung der Preis bestimmt werden. Dies ermöglicht die Berechnung von preisoptimalen Routen. An den Tarifsystemen der Verkehrsverbünde MDV (Mitteldeutscher Verkehrsverbund) und VBB (Verkehrsverbund Berlin-Brandenburg) wird die Konstruktion des Modells detailliert erläutert. Y1 - 2021 UR - https://verlag.fgsv-datenbanken.de/tagungsbaende?kat=HEUREKA&subkat=FGSV+002%2F127+%282021%29&fanr=&va=&titel=&text=&autor=&tagungsband=1256&_titel=Ein+Graphen-basiertes+Modell+zur+Beschreibung+von+Preissystemen+im+%C3%B6ffentlichen+Nahverkehr VL - 002/127 SP - 1 EP - 15 PB - FGSV CY - Stuttgart ER - TY - JOUR A1 - Prause, Felix A1 - Borndörfer, Ralf T1 - An Iterative Refinement Approach for the Rolling Stock Rotation Problem with Predictive Maintenance N2 - The rolling stock rotation problem with predictive maintenance (RSRP-PdM) involves the assignment of trips to a fleet of vehicles with integrated maintenance scheduling based on the predicted failure probability of the vehicles. These probabilities are determined by the health states of the vehicles, which are considered to be random variables distributed by a parameterized family of probability distribution functions. During the operation of the trips, the corresponding parameters get updated. In this article, we present a dual solution approach for RSRP-PdM and generalize a linear programming based lower bound for this problem to families of probability distribution functions with more than one parameter. For this purpose, we define a rounding function that allows for a consistent underestimation of the parameters and model the problem by a state-expanded event-graph in which the possible states are restricted to a discrete set. This induces a flow problem that is solved by an integer linear program. We show that the iterative refinement of the underlying discretization leads to solutions that converge from below to an optimal solution of the original instance. Thus, the linear relaxation of the considered integer linear program results in a lower bound for RSRP-PdM. Finally, we report on the results of computational experiments conducted on a library of test instances. Y1 - 2024 ER - TY - JOUR A1 - Prause, Felix A1 - Borndörfer, Ralf A1 - Grimm, Boris A1 - Tesch, Alexander T1 - Approximating rolling stock rotations with integrated predictive maintenance JF - Journal of Rail Transport Planning & Management N2 - We study the solution of the rolling stock rotation problem with predictive maintenance (RSRP-PdM) by an iterative refinement approach that is based on a state-expanded event-graph. In this graph, the states are parameters of a failure distribution, and paths correspond to vehicle rotations with associated health state approximations. An optimal set of paths including maintenance can be computed by solving an integer linear program. Afterwards, the graph is refined and the procedure repeated. An associated linear program gives rise to a lower bound that can be used to determine the solution quality. Computational results for six instances derived from real-world timetables of a German railway company are presented. The results show the effectiveness of the approach and the quality of the solutions. Y1 - 2024 U6 - https://doi.org/10.1016/j.jrtpm.2024.100434 VL - 30 SP - 100434 ER - TY - JOUR A1 - Sahin, Guvenc A1 - Digehsara, Amin Ahmadi A1 - Borndörfer, Ralf T1 - Efficient Algorithms for the Multi-Period Line Planning Problem in Public Transportation JF - Open Access Series in Informatics (OASIcs), 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021) N2 - In order to plan and schedule a demand-responsive public transportation system, both temporal and spatial changes in demand should be taken into account even at the line planning stage. We study the multi-period line planning problem with integrated decisions regarding dynamic allocation of vehicles among the lines. Given the NP-hard nature of the line planning problem, the multi-period version is clearly difficult to solve for large public transit networks even with advanced solvers. It becomes necessary to develop algorithms that are capable of solving even the very-large instances in reasonable time. For instances which belong to real public transit networks, we present results of a heuristic local branching algorithm and an exact approach based on constraint propagation. Y1 - 2021 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2021.17 VL - 96 SP - 17:1 EP - 17:6 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - A Discrete-Continuous Algorithm for Free Flight Planning JF - Algorithms N2 - We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach. KW - shortest path KW - flight planning KW - free flight KW - discrete-continuous algorithm KW - optimal control KW - discrete optimization Y1 - 2020 U6 - https://doi.org/10.3390/a14010004 SN - 1438-0064 VL - 14 IS - 1 SP - 4 PB - MDPI ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Eßer, Thomas A1 - Frankenberger, Patrick A1 - Huck, Andreas A1 - Jobmann, Christoph A1 - Krostitz, Boris A1 - Kuchenbecker, Karsten A1 - Moorhagen, Kai A1 - Nagl, Philipp A1 - Peterson, Michael A1 - Reuther, Markus A1 - Schang, Thilo A1 - Schoch, Michael A1 - Schülldorf, Hanno A1 - Schütz, Peter A1 - Therolf, Tobias A1 - Waas, Kerstin A1 - Weider, Steffen T1 - Deutsche Bahn Schedules Train Rotations Using Hypergraph Optimization JF - Informs Journal on Applied Analytics N2 - Deutsche Bahn (DB) operates a large fleet of rolling stock (locomotives, wagons, and train sets) that must be combined into trains to perform rolling stock rotations. This train composition is a special characteristic of railway operations that distinguishes rolling stock rotation planning from the vehicle scheduling problems prevalent in other industries. DB models train compositions using hyperarcs. The resulting hypergraph models are ad-dressed using a novel coarse-to-fine method that implements a hierarchical column genera-tion over three levels of detail. This algorithm is the mathematical core of DB’s fleet em-ployment optimization (FEO) system for rolling stock rotation planning. FEO’s impact within DB’s planning departments has been revolutionary. DB has used it to support the company’s procurements of its newest high-speed passenger train fleet and its intermodal cargo locomotive fleet for cross-border operations. FEO is the key to successful tendering in regional transport and to construction site management in daily operations. DB’s plan-ning departments appreciate FEO’s high-quality results, ability to reoptimize (quickly), and ease of use. Both employees and customers benefit from the increased regularity of operations. DB attributes annual savings of 74 million euro, an annual reduction of 34,000 tons of CO2 emissions, and the elimination of 600 coupling operations in cross-border operations to the implementation of FEO. Y1 - 2021 U6 - https://doi.org/10.1287/inte.2020.1069 VL - 51 IS - 1 SP - 42 EP - 62 ER - TY - JOUR A1 - Sahin, Guvenc A1 - Ahmadi, Amin A1 - Borndörfer, Ralf A1 - Schlechte, Thomas T1 - Multi-period line planning with resource transfers JF - Transportation Research Part C: Emerging Technologies N2 - Urban transportation systems are subject to a high level of variation and fluctuation in demand over the day. When this variation and fluctuation are observed in both time and space, it is crucial to develop line plans that are responsive to demand. A multi-period line planning approach that considers a changing demand during the planning horizon is proposed. If such systems are also subject to limitations of resources, a dynamic transfer of resources from one line to another throughout the planning horizon should also be considered. A mathematical modelling framework is developed to solve the line planning problem with a cost-oriented approach considering transfer of resources during a finite length planning horizon of multiple periods. We use real-life public transportation network data for our computational results. We analyze whether or not multi-period solutions outperform single period solutions in terms of feasibility and relevant costs. The importance of demand variation on multi-period solutions is investigated. We evaluate the impact of resource transfer constraints on the effectiveness of solutions. We also study the effect of period lengths along with the problem parameters that are significant for and sensitive to the optimality of solutions. Y1 - 2020 U6 - https://doi.org/10.1016/j.trc.2020.102726 VL - 119 SP - 102726 ER - TY - JOUR A1 - Euler, Ricardo A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - Price optimal routing in public transportation JF - EURO Journal on Transportation and Logistics KW - Management Science and Operations Research KW - Transportation KW - Modeling and Simulation Y1 - 2024 U6 - https://doi.org/10.1016/j.ejtl.2024.100128 SN - 2192-4376 VL - 13 SP - 1 EP - 15 PB - Elsevier BV ER - TY - JOUR A1 - Euler, Ricardo A1 - Maristany de las Casas, Pedro A1 - Borndörfer, Ralf T1 - Logic-Constrained Shortest Paths for Flight Planning N2 - The Logic-Constrained Shortest Path Problem (LCSP) combines a one-to-one shortest path problem with satisfiability constraints imposed on the routing graph. This setting arises in flight planning, where air traffic control (ATC) authorities are enforcing a set of traffic flow restrictions (TFRs) on aircraft routes in order to increase safety and throughput. We propose a new branch and bound-based algorithm for the LCSP. The resulting algorithm has three main degrees of freedom: the node selection rule, the branching rule and the conflict. While node selection and branching rules have been long studied in the MIP and SAT communities, most of them cannot be applied out of the box for the LCSP. We review the existing literature and develop tailored variants of the most prominent rules. The conflict, the set of variables to which the branching rule is applied, is unique to the LCSP. We analyze its theoretical impact on the B&B algorithm. In the second part of the paper, we show how to model the Flight Planning Problem with TFRs as an LCSP and solve it using the branch and bound algorithm. We demonstrate the algorithm’s efficiency on a dataset consisting of a global flight graph and a set of around 20000 real TFRs obtained from our industry partner Lufthansa Systems GmbH. We make this dataset publicly available. Finally, we conduct an empirical in-depth analysis of node selection rules, branching rules and conflicts. Carefully choosing an appropriate combination yields an improvement of an order of magnitude compared to an uninformed choice. Y1 - 2024 ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Ebert, Patricia T1 - Forward and Line-Based Cycle Bases for Periodic Timetabling JF - Operations Research Forum N2 - The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances. Y1 - 2023 U6 - https://doi.org/10.1007/s43069-023-00229-0 VL - 4 IS - 3 SP - 53 ER - TY - JOUR A1 - Maristany de las Casas, Pedro A1 - Sedeno-Noda, Antonio A1 - Borndörfer, Ralf T1 - An Improved Multiobjective Shortest Path Algorithm JF - Computers & Operations Research N2 - We present a new label-setting algorithm for the Multiobjective Shortest Path (MOSP) problem that computes a minimum complete set of efficient paths for a given instance. The size of the priority queue used in the algorithm is bounded by the number of nodes in the input graph and extracted labels are guaranteed to be efficient. These properties allow us to give a tight output-sensitive running time bound for the new algorithm that can almost be expressed in terms of the running time of Dijkstra’s algorithm for the Shortest Path problem. Hence, we suggest to call the algorithm Multiobjective Dijkstra Algorithm (MDA). The simplified label management in the MDA allows us to parallelize some subroutines. In our computational experiments, we compare the MDA and the classical label-setting MOSP algorithm by Martins, which we improved using new data structures and pruning techniques. On average, the MDA is 2 to 9 times faster on all used graph types. On some instances the speedup reaches an order of magnitude. Y1 - 2021 U6 - https://doi.org/10.1016/j.cor.2021.105424 VL - 135 ER - TY - JOUR A1 - Schlechte, Thomas A1 - Borndörfer, Ralf A1 - Denißen, Jonas A1 - Heller, Simon A1 - Klug, Torsten A1 - Küpper, Michael A1 - Lindner, Niels A1 - Reuther, Markus A1 - Söhlke, Andreas A1 - Steadman, William T1 - Timetable Optimization for a Moving Block System JF - Journal of Rail Transport Planning & Management N2 - We present an optimization model which is capable of routing and ordering trains on a microscopic level under a moving block regime. Based on a general timetabling definition (GTTP) that allows the plug in of arbitrarily detailed methods to compute running and headway times, we describe a layered graph approach using velocity expansion, and develop a mixed integer linear programming formulation. Finally, we present promising results for a German corridor scenario with mixed traffic, indicating that applying branch-and-cut to our model is able to solve reasonably sized instances with up to hundred trains to optimality. Y1 - 2022 U6 - https://doi.org/10.1016/j.jrtpm.2022.100315 SN - 2210-9706 VL - 22 SP - 100315 ER -