TY - GEN A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - The tropical and zonotopal geometry of periodic timetables N2 - The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetabling problems in public transport. A solution to PESP consists of three parts: a periodic timetable, a periodic tension, and integer periodic offset values. While the space of periodic tension has received much attention in the past, we explore geometric properties of the other two components, establishing novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables, and decompose it into polytropes, i.e., 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 the tropical neighbourhood of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope. We relate its zonotopal tilings back to the hyperrectangle of fractional periodic tensions and to the tropical neighbourhood of the periodic timetable space. To conclude we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis. T3 - ZIB-Report - 22-09 KW - periodic event scheduling KW - tropical geometry KW - zonotopal tilings Y1 - 2022 SN - 1438-0064 ER - TY - CHAP A1 - Lindner, Niels A1 - Liebchen, Christian A1 - Masing, Berenike T1 - Forward Cycle Bases and Periodic Timetabling T2 - 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021) N2 - Periodic timetable optimization problems in public transport can be modeled as mixed-integer linear programs by means of the Periodic Event Scheduling Problem (PESP). In order to keep the branch-and-bound tree small, minimum integral cycle bases have been proven successful. We examine forward cycle bases, where no cycle is allowed to contain a backward arc. After reviewing the theory of these bases, we describe the construction of an integral forward cycle basis on a line-based event-activity network. Adding turnarounds to the instance R1L1 of the benchmark library PESPlib, we computationally evaluate three types of forward cycle bases in the Pareto sense, and come up with significant improvements concerning dual bounds. Y1 - 2021 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2021.2 VL - 96 SP - 2:1 EP - 2:14 ER - TY - CHAP A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling T2 - 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022) N2 - Periodic timetabling is a central aspect of both the long-term organization and the day-to-day operations of a public transportation system. The Periodic Event Scheduling Problem (PESP), the combinatorial optimization problem that forms the mathematical basis of periodic timetabling, is an extremely hard problem, for which optimal solutions are hardly ever found in practice. The most prominent solving strategies today are based on mixed-integer programming, and there is a concurrent PESP solver employing a wide range of heuristics [Borndörfer et al., 2020]. We present tropical neighborhood search (tns), a novel PESP heuristic. The method is based on the relations between periodic timetabling and tropical geometry [Bortoletto et al., 2022]. We implement tns into the concurrent solver, and test it on instances of the benchmarking library PESPlib. The inclusion of tns turns out to be quite beneficial to the solver: tns is able to escape local optima for the modulo network simplex algorithm, and the overall share of improvement coming from tns is substantial compared to the other methods available in the solver. Finally, we provide better primal bounds for five PESPlib instances. Y1 - 2022 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2022.3 VL - 106 SP - 3:1 EP - 3:19 ER - TY - GEN A1 - Masing, Berenike A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - The Price of Symmetric Line Plans in the Parametric City N2 - We consider the line planning problem in public transport in the Parametric City, an idealized model that captures typical scenarios by a (small) number of parameters. The Parametric City is rotation symmetric, but optimal line plans are not always symmetric. This raises the question to quantify the symmetry gap between the best symmetric and the overall best solution. For our analysis, we formulate the line planning problem as a mixed integer linear program, that can be solved in polynomial time if the solutions are forced to be symmetric. The symmetry gap is provably small when a specific Parametric City parameter is fixed, and we give an approximation algorithm for line planning in the Parametric City in this case. While the symmetry gap can be arbitrarily large in general, we show that symmetric line plans are a good choice in most practical situations. T3 - ZIB-Report - 22-01 KW - line planning KW - mixed integer programming KW - parametric city modeling Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-85648 SN - 1438-0064 ER - TY - GEN A1 - Lindner, Niels A1 - Liebchen, Christian A1 - Masing, Berenike T1 - Forward Cycle Bases and Periodic Timetabling N2 - Periodic timetable optimization problems in public transport can be modeled as mixed-integer linear programs by means of the Periodic Event Scheduling Problem (PESP). In order to keep the branch-and-bound tree small, minimum integral cycle bases have been proven successful. We examine forward cycle bases, where no cycle is allowed to contain a backward arc. After reviewing the theory of these bases, we describe the construction of an integral forward cycle basis on a line-based event-activity network. Adding turnarounds to the instance \texttt{R1L1} of the benchmark library PESPlib, we computationally evaluate three types of forward cycle bases in the Pareto sense, and come up with significant improvements concerning dual bounds. T3 - ZIB-Report - 21-18 KW - Periodic Timetabling KW - Cycle Bases KW - Mixed Integer Programming Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-82756 SN - 1438-0064 ER - TY - THES A1 - Masing, Berenike T1 - Optimal Line Planning in the Parametric City N2 - One of the fundamental steps in the optimization of public transport is line planning. It involves determining lines and assigning frequencies of service such that costs are minimized while also maximizing passenger comfort and satisfying travel demands. We formulate the problem as a mixed integer linear program that considers all circuit-like lines in a graph and allows free passenger routing. Traveler and operator costs are included in a linear scalarization in the objective. We apply said programming problem to the Parametric City, which is a graph model introduced by Fielbaum, Jara-Díaz and Gschwender that exibly represents different cities. In his dissertation, Fielbaum solved the line planning problem for various parameter choices in the Parametric City. In a first step, we therefore review his results and make comparative computations. Unlike Fielbaum we arrive at the conclusion that the optimal line plan for this model indeed depends on the demand. Consequently, we analyze the line planning problem in-depth: We find equivalent, but easier to compute formulations and provide a lower bound by LP-relaxation, which we show to be equivalent to a multi-commodity flow problem. Further, we examine what impact symmetry has on the solutions. Supported both by computational results as well as by theoretical analysis, we reach the conclusion that symmetric line plans are optimal or near-optimal in the Parametric City. Restricting the model to symmetric line plans allows for a \kappa-factor approximation algorithm for the line planning problem in the Parametric City. Y1 - 2020 ER - TY - GEN A1 - Bortoletto, Enrico A1 - Lindner, Niels A1 - Masing, Berenike T1 - Tropical Neighbourhood Search: A New Heuristic for Periodic Timetabling N2 - Periodic timetabling is a central aspect of both the long-term organization and the day-to-day operations of a public transportation system. The Periodic Event Scheduling Problem (PESP), the combinatorial optimization problem that forms the mathematical basis of periodic timetabling, is an extremely hard problem, for which optimal solutions are hardly ever found in practice. The most prominent solving strategies today are based on mixed-integer programming, and there is a concurrent PESP solver employing a wide range of heuristics [3]. We present tropical neighborhood search (tns), a novel PESP heuristic. The method is based on the relations between periodic timetabling and tropical geometry [4]. We implement tns into the concurrent solver, and test it on instances of the benchmarking library PESPlib. The inclusion of tns turns out to be quite beneficial to the solver: tns is able to escape local optima for the modulo network simplex algorithm, and the overall share of improvement coming from tns is substantial compared to the other methods available in the solver. Finally, we provide better primal bounds for five PESPlib instances. T3 - ZIB-Report - 22-13 KW - Periodic Timetabling KW - Tropical Geometry KW - Neighbourhood Search KW - Mixed-Integer Programming Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-87385 SN - 1438-0064 ER - TY - CHAP A1 - Masing, Berenike ED - Trautmann, Norbert ED - Drägi, Mario T1 - Optimal Line Planning in the Parametric City T2 - Operations Research Proceedings 2021. OR 2021. N2 - We formulate the line planning problem in public transport as a mixed integer linear program (MILP), which selects both passenger and vehicle routes, such that travel demands are met with respect to minimized travel times for both operators and users. We apply MILP to the Parametric City, a generic city model developed by Fielbaum et al. [2]. While the infrastructure graph and demand are entirely rotation symmetric, asymmetric optimal line plans can occur. Using group theory, we analyze the properties of symmetric solutions and introduce a symmetry gap to measure their deviation of the optimum. We also develop a 1+1+2√g-approximation algorithm, depending only on the cost related parameter g. Supported by computational experiments, we conclude that in practice symmetric line plans provide good solutions for the line planning problem in the Parametric City. Y1 - 2022 SN - 978-3-031-08623-6 U6 - https://doi.org/10.1007/978-3-031-08623-6_7 SP - 39 EP - 44 PB - Springer, Cham ER - TY - JOUR A1 - Masing, Berenike A1 - Lindner, Niels A1 - Borndörfer, Ralf T1 - The price of symmetric line plans in the Parametric City JF - Transportation Research Part B: Methodological N2 - We consider the line planning problem in public transport in the Parametric City, an idealized model that captures typical scenarios by a (small) number of parameters. The Parametric City is rotation symmetric, but optimal line plans are not always symmetric. This raises the question to quantify the symmetry gap between the best symmetric and the overall best solution. For our analysis, we formulate the line planning problem as a mixed integer linear program, that can be solved in polynomial time if the solutions are forced to be symmetric. We prove that the symmetry gap is small when a specific Parametric City parameter is fixed, and we give an approximation algorithm for line planning in the Parametric City in this case. While the symmetry gap can be arbitrarily large in general, we show that symmetric line plans are a good choice in most practical situations. Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-88613 VL - 166 SP - 419 EP - 443 ER - TY - GEN A1 - Masing, Berenike T1 - Optimal Line Planning in the Parametric City N2 - We formulate the line planning problem in public transport as a mixed integer linear program (MILP), which selects both passenger and vehicle routes, such that travel demands are met with respect to minimized travel times for both operators and users. We apply MILP to the Parametric City, a generic city model developed by Fielbaum et al. While the infrastructure graph and demand are entirely rotation symmetric, asymmetric optimal line plans can occur. Using group theory, we analyze the properties of symmetric solutions and introduce a symmetry gap to measure their deviation of the optimum. We also develop a 1+(1+\sqrt{2})/g-approximation algorithm, depending only on the cost related parameter g. Supported by computational experiments, we conclude that in practice symmetric line plans provide good solutions for the line planning problem in the Parametric City. T3 - ZIB-Report - 21-26 KW - line planning KW - city modelling KW - symmetry KW - mixed integer programming KW - approximation algorithm Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-83238 SN - 1438-0064 ER -