@inproceedings{BorndoerferKarbsteinLiebchenetal.2018, author = {Bornd{\"o}rfer, Ralf and Karbstein, Marika and Liebchen, Christian and Lindner, Niels}, title = {A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable}, volume = {65}, booktitle = {18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018)}, doi = {10.4230/OASIcs.ATMOS.2018.16}, pages = {16:1 -- 16:15}, year = {2018}, abstract = {We consider the following planning problem in public transportation: Given a periodic timetable, how many vehicles are required to operate it? In [9], for this sequential approach, it is proposed to first expand the periodic timetable over time, and then answer the above question by solving a flow-based aperiodic optimization problem. In this contribution we propose to keep the compact periodic representation of the timetable and simply solve a particular perfect matching problem. For practical networks, it is very much likely that the matching problem decomposes into several connected components. Our key observation is that there is no need to change any turnaround decision for the vehicles of a line during the day, as long as the timetable stays exactly the same.}, language = {en} } @misc{BorndoerferHoppmannKarbsteinetal.2018, author = {Bornd{\"o}rfer, Ralf and Hoppmann, Heide and Karbstein, Marika and Lindner, Niels}, title = {Separation of Cycle Inequalities in Periodic Timetabling}, issn = {1438-0064}, doi = {https://doi.org/10.1016/j.disopt.2019.100552}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69746}, year = {2018}, abstract = {Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem. We give the first pseudo-polynomial time separation algo- rithm for cycle inequalities, and we give 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.}, language = {en} } @article{RiedmuellerLindner2025, author = {Riedm{\"u}ller, Stephanie and Lindner, Niels}, title = {Column Generation for Periodic Timetabling}, arxiv = {http://arxiv.org/abs/2510.12466}, year = {2025}, abstract = {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.}, language = {en} } @inproceedings{LoebelLindner2025, author = {L{\"o}bel, Fabian and Lindner, Niels}, title = {A Geometric Approach to Integrated Periodic Timetabling and Passenger Routing}, volume = {137}, booktitle = {25th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2025)}, editor = {Sauer, Jonas and Schmidt, Marie}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, isbn = {978-3-95977-404-8}, issn = {2190-6807}, doi = {10.4230/OASIcs.ATMOS.2025.2}, url = {http://nbn-resolving.de/urn:nbn:de:0030-drops-247580}, pages = {2:1 -- 2:19}, year = {2025}, abstract = {We offer a geometric perspective on the problem of integrated periodic timetabling and passenger routing in public transport. Inside the space of periodic tensions, we single out those regions, where the same set of paths provides shortest passenger routes. This results in a polyhedral subdivision, which we combine with the known decomposition by polytropes. On each maximal region of the common refinement, the integrated problem is solvable in polynomial time. We transform these insights into a new geometry-driven primal heuristic, integrated tropical neighborhood search (ITNS). Computationally, we compare implementations of ITNS and the integrated (restricted) modulo network simplex algorithm on the TimPassLib benchmark set, and contribute better solutions in terms of total travel time for all but one of the twenty-five instances for which a proven optimal solution is not yet known.}, language = {en} } @article{vanLieshoutLindner2025, author = {van Lieshout, Rolf and Lindner, Niels}, title = {A Compact Cycle Formulation for the Multiperiodic Event Scheduling Problem}, arxiv = {http://arxiv.org/abs/2507.17566}, year = {2025}, abstract = {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.}, language = {en} } @article{MasingLindnerBortoletto2025, author = {Masing, Berenike and Lindner, Niels and Bortoletto, Enrico}, title = {Computing All Shortest Passenger Routes with a Tropical Dijkstra Algorithm}, volume = {14}, journal = {EURO Journal on Transportation and Logistics}, arxiv = {http://arxiv.org/abs/2412.14654}, doi = {10.1016/j.ejtl.2025.100163}, year = {2025}, abstract = {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.}, language = {en} } @article{BortolettoLindnerMasing2025, author = {Bortoletto, Enrico and Lindner, Niels and Masing, Berenike}, title = {The Tropical and Zonotopal Geometry of Periodic Timetables}, volume = {73}, journal = {Discrete \& Computational Geometry}, doi = {10.1007/s00454-024-00686-2}, pages = {719 -- 763}, year = {2025}, abstract = {The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetables in public transport. A solution to a PESP instance consists of three parts: a periodic timetable, a periodic tension, and integer offset values. While the space of periodic tensions has received much attention in the past, we explore geometric properties of the other two components. The general aim of this paper is to establish novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables as a disjoint union of polytropes. These are polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on neighbourhood relations of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope, and then study its zonotopal tilings. These are related to the hyperrectangle of fractional periodic tensions, as well as the polytropes of the periodic timetable space, and we detail their interplay. To conclude, we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.}, language = {en} } @inproceedings{LindnerLiebchen2025, author = {Lindner, Niels and Liebchen, Christian}, title = {Incremental Heuristics for Periodic Timetabling}, booktitle = {Operations Research Proceedings 2023}, editor = {Voigt, Guido and Fliedner, Malte and Haase, Knut and Br{\"u}ggemann, Wolfgang and Hoberg, Kai and Meissner, J{\"o}rn}, publisher = {Springer Nature Switzerland}, address = {Cham}, doi = {10.1007/978-3-031-58405-3_59}, pages = {461 -- 467}, year = {2025}, abstract = {We present incremental heuristics for the Periodic Event Scheduling Problem (PESP), the standard mathematical tool to optimize periodic timetables in public transport. The core of our method is to solve successively larger subinstances making use of previously found solutions. Introducing the technical notion of free stratifications, we formulate a general scheme for incremental heuristics for PESP. More practically, we use line and station information to create heuristics that add lines or stations one by one, and we evaluate these heuristics on instances of the benchmarking library PESPlib. This approach is indeed viable, and leads to new incumbent solutions for six PESPlib instances.}, language = {en} } @misc{LindnerMasing2024, author = {Lindner, Niels and Masing, Berenike}, title = {SAT-Generated Initial Solutions for Integrated Line Planning and Turn-Sensitive Periodic Timetabling with Track Choice}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-94644}, year = {2024}, abstract = {Periodic timetabling is a challenging planning task in public transport. As safety requirements are crucial, track allocation is indispensable for validating the practical feasibility of a railway timetable. For busy stations with limited capacities, this requires a detailed planning of turnarounds. It is therefore desirable to integrate timetabling not only with track allocation, but also with vehicle scheduling and line planning. This is captured by the Integrated Line Planning and Turn-Sensitive Periodic Timetabling Problem with Track Choice, whose MIP formulation has been demonstrated to be effective for construction site railway rescheduling, as long as a good quality initial solution is available. In this paper, we discuss how to generate such a solution by extending the SAT formulation of the Periodic Event Scheduling Problem with track choice, track occupation, and minimum service frequency components. The SAT approach is superior to pure MIP on real-world instances of the S-Bahn Berlin network.}, language = {en} } @misc{BorndoerferDenissenHelleretal.2021, author = {Bornd{\"o}rfer, Ralf and Denißen, Jonas and Heller, Simon and Klug, Torsten and K{\"u}pper, Michael and Lindner, Niels and Reuther, Markus and Schlechte, Thomas and S{\"o}hlke, Andreas and Steadman, William}, title = {Microscopic Timetable Optimization for a Moving Block System}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82547}, year = {2021}, abstract = {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.}, language = {en} } @misc{LindnerLiebchen2020, author = {Lindner, Niels and Liebchen, Christian}, title = {Determining all integer vertices of the PESP polytope by flipping arcs}, issn = {1438-0064}, doi = {10.4230/OASIcs.ATMOS.2020.5}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78793}, year = {2020}, abstract = {We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. These produce better dual bounds for the smallest and largest instance of the benchmarking library PESPlib.}, language = {en} } @misc{LindnerMasing2023, author = {Lindner, Niels and Masing, Berenike}, title = {On the Split Closure of the Periodic Timetabling Polytope}, issn = {1438-0064}, doi = {10.1007/s10107-025-02220-5}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-91156}, year = {2023}, abstract = {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.}, language = {en} } @misc{LindnerLiebchen2023, author = {Lindner, Niels and Liebchen, Christian}, title = {Incremental Heuristics for Periodic Timetabling}, issn = {1438-0064}, doi = {10.1007/978-3-031-58405-3_59}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-92309}, year = {2023}, abstract = {We present incremental heuristics for the Periodic Event Scheduling Problem (PESP), the standard mathematical tool to optimize periodic timetables in public transport. The core of our method is to solve successively larger subinstances making use of previously found solutions. Introducing the technical notion of free stratifications, we formulate a general scheme for incremental heuristics for PESP. More practically, we use line and station information to create heuristics that add lines or stations one by one, and we evaluate these heuristics on instances of the benchmarking library PESPlib. This approach is indeed viable, and leads to new incumbent solutions for six PESPlib instances.}, language = {en} } @misc{BortolettoLindner2023, author = {Bortoletto, Enrico and Lindner, Niels}, title = {Scaling and Rounding Periodic Event Scheduling Instances to Different Period Times}, issn = {1438-0064}, doi = {10.1007/978-3-031-58405-3_51}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-92315}, year = {2023}, abstract = {The Periodic Event Scheduling Problem (PESP) is a notoriously hard combinatorial optimization problem, essential for the design of periodic timetables in public transportation. The coefficients of the integer variables in the standard mixed integer linear programming formulations of PESP are the period time, e.g., 60 for a horizon of one hour with a resolution of one minute. In many application scenarios, lines with different frequencies have to be scheduled, leading to period times with many divisors. It then seems natural to consider derived instances, where the period time is a divisor of the original one, thereby smaller, and bounds are scaled and rounded accordingly. To this end, we identify two rounding schemes: wide and tight. We then discuss the approximation performance of both strategies, in theory and practice.}, language = {en} } @misc{MasingLindnerEbert2023, author = {Masing, Berenike and Lindner, Niels and Ebert, Patricia}, title = {Forward and Line-Based Cycle Bases for Periodic Timetabling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-89731}, year = {2023}, abstract = {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.}, language = {en} } @article{LindnerMasing2025, author = {Lindner, Niels and Masing, Berenike}, title = {On the split closure of the periodic timetabling polytope}, journal = {Mathematical Programming}, publisher = {Springer Science and Business Media LLC}, issn = {0025-5610}, doi = {10.1007/s10107-025-02220-5}, year = {2025}, abstract = {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.}, language = {en} } @misc{LindnerMehlBartoszuketal.2025, author = {Lindner, Niels and Mehl, Lukas and Bartoszuk, Karolina and Berendes, Sarah and Zittel, Janina}, title = {Demand Uncertainty in Energy Systems: Scenario Catalogs vs. Integrated Robust Optimization}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-102404}, year = {2025}, abstract = {Designing efficient energy systems is indispensable for shaping a more sustainable society. This involves making infrastructure investment decisions that must be valid for a long-term time horizon. While energy system optimization models constitute a powerful technique to support planning decisions, they need to cope with inherent uncertainty. For example, predicting future demand on a scale of decades is not only an intricate challenge in itself, but small fluctuations in such a forecast might also largely impact the layout of a complex energy system. In this paper, we compare two methodologies of capturing demand uncertainty for linear-programming based energy system optimization models. On one hand, we generate and analyze catalogs of varying demand scenarios, where each individual scenario is considered independently, so that the optimization produces scenario-specific investment pathways. On the other hand, we make use of robust linear programming to meet the demand of all scenarios at once. Since including a multitude of scenarios increases the size and complexity of the optimization model, we will show how to use warm-starting approaches to accelerate the computation process, by exploiting the similar structure of the linear program across different demand inputs. This allows to integrate a meaningful number of demand scenarios with fully-fledged energy system models. We demonstrate the practical use of our methods in a case study of the Berlin-Brandenburg area in Germany, a region that contains both a metropolitan area and its rural surroundings. As a backbone, we use the open-source framework oemof to create a sector-coupled optimization model for planning an energy system with up to 100\% reduction of greenhouse gas emissions. This model features a fine-grained temporal resolution of one hour for the full year 2050. We consider uncertainty in demand for electricity, hydrogen, natural gas, central, and decentral heat. Based on our computations, we analyze the trade-offs in terms of quality and computation time for scenario catalogs and the robust optimization approach. We further demonstrate that our procedure provides a valuable strategy for decision makers to gain insight on the robustness and sensitivity of solutions regarding demand variability.}, language = {en} } @misc{Lindner2025, author = {Lindner, Niels}, title = {A Multi-Commodity Flow Heuristic for Integrated Periodic Timetabling for Railway Construction Sites}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-99449}, year = {2025}, abstract = {Rescheduling a railway system comprises many aspects, such as line planning, timetabling, track allocation, and vehicle scheduling. For periodic timetables, these features can be integrated into a single mixed-integer program extending the Periodic Event Scheduling Problem (PESP) with a routing component. We develop a multi-commodity-flow-based heuristic that allows to compute better solutions faster than a black-box MIP approach on real construction site scenarios on the S-Bahn Berlin network.}, language = {en} } @misc{LindnerLiebchen2025, author = {Lindner, Niels and Liebchen, Christian}, title = {A two-stage model for periodic timetabling with fixed line activities}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-101234}, year = {2025}, abstract = {The timetable is a central pillar of any public transportation system. Constructing and optimizing periodic timetables in terms of passenger comfort and operational efficiency leads to NP-hard optimization problems that are also computationally challenging in applications. The Periodic Event Scheduling Problem (PESP) as standard mathematical tool benefits from its succinct formulation and rich combinatorial structure, but suffers from poor linear programming relaxations and weak dual bounds. These difficulties persist in a reduced version, where driving and dwelling activities of the lines are assumed to be fixed. In this case, fixing the initial departure time of each line fully determines the timetable, and for each pair of lines, the resulting (weighted) transfer durations can be expressed in terms of a piecewise linear non-convex function in terms of the difference of the initial times. When the number of activities between two lines is bounded, this function can be computed in polynomial time. By inserting precomputed piecewise linear functions into a mixed-integer program with the initial departure times as variables, we introduce an equivalent formulation for reduced PESP instances. The model bears analogies with quadratic semi-assignment approaches and offers alternative ways to compute primal and dual bounds. We evaluate the computational behavior of our approach on realistic benchmarking instances.}, language = {en} } @inproceedings{LindnerMehlBartoszuketal.2025, author = {Lindner, Niels and Mehl, Lukas and Bartoszuk, Karolina and Berendes, Sarah and Zittel, Janina}, title = {Demand Uncertainty in Energy Systems: Scenario Catalogs vs. Integrated Robust Optimization}, booktitle = {Proceedings of the 38th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems}, year = {2025}, abstract = {Designing efficient energy systems is indispensable for shaping a more sustainable society. This involves making infrastructure investment decisions that must be valid for a long-term time horizon. While energy system optimization models constitute a powerful technique to support planning decisions, they need to cope with inherent uncertainty. For example, predicting future demand on a scale of decades is not only an intricate challenge in itself, but small fluctuations in such a forecast might also largely impact the layout of a complex energy system. In this paper, we compare two methodologies of capturing demand uncertainty for linear-programming based energy system optimization models. On one hand, we generate and analyze catalogs of varying demand scenarios, where each individual scenario is considered independently, so that the optimization produces scenario-specific investment pathways. On the other hand, we make use of robust linear programming to meet the demand of all scenarios at once. Since including a multitude of scenarios increases the size and complexity of the optimization model, we will show how to use warm-starting approaches to accelerate the computation process, by exploiting the similar structure of the linear program across different demand inputs. This allows to integrate a meaningful number of demand scenarios with fully-fledged energy system models. We demonstrate the practical use of our methods in a case study of the Berlin-Brandenburg area in Germany, a region that contains both a metropolitan area and its rural surroundings. As a backbone, we use the open-source framework oemof to create a sector-coupled optimization model for planning an energy system with up to 100\% reduction of greenhouse gas emissions. This model features a fine-grained temporal resolution of one hour for the full year 2050. We consider uncertainty in demand for electricity, hydrogen, natural gas, central, and decentral heat. Based on our computations, we analyze the trade-offs in terms of quality and computation time for scenario catalogs and the robust optimization approach. We further demonstrate that our procedure provides a valuable strategy for decision makers to gain insight on the robustness and sensitivity of solutions regarding demand variability.}, language = {en} } @misc{LindnerLiebchen2021, author = {Lindner, Niels and Liebchen, Christian}, title = {Timetable Merging for the Periodic Event Scheduling Problem}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81587}, year = {2021}, abstract = {We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.}, language = {en} }