TY - GEN A1 - Schwartz, Stephan T1 - An Overview of Graph Covering and Partitioning N2 - While graph covering is a fundamental and well-studied problem, this field lacks a broad and unified literature review. The holistic overview of graph covering given in this article attempts to close this gap. The focus lies on a characterization and classification of the different problems discussed in the literature. In addition, notable results and common approaches are also included. Whenever appropriate, our review extends to the corresponding partioning problems. T3 - ZIB-Report - 20-24 KW - Graph Covering KW - Graph Partitioning Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-79431 SN - 1438-0064 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Danecker, Fabian A1 - Weiser, Martin T1 - A Discrete-Continuous Algorithm for Free Flight Planning 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. T3 - ZIB-Report - 20-33 KW - shortest path KW - flight planning KW - free flight KW - discrete-continuous algorithm KW - optimal control KW - discrete optimization Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-81343 SN - 1438-0064 ER - TY - GEN A1 - Maristany de las Casas, Pedro A1 - Borndörfer, Ralf A1 - Kraus, Luitgard A1 - Sedeño-Noda, Antonio T1 - An FPTAS for Dynamic Multiobjective Shortest Path Problems N2 - We propose in this paper the Dynamic Multiobjective Shortest Problem. It features multidimensional states that can depend on several variables and not only on time; this setting is motivated by flight planning and electric vehicle routing applications. We give an exact algorithm for the FIFO case and derive from it an FPTAS, which is computationally efficient. It also features the best known complexity in the static case. T3 - ZIB-Report - 20-31 KW - Multiobjective Shortest Paths KW - Time Dependent Shortest Paths KW - Multiobjective Approximation Algorithms KW - Flight Planning Problem Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-80954 ER - TY - JOUR A1 - Schienle, Adam A1 - Maristany de las Casas, Pedro A1 - Blanco, Marco T1 - A Priori Search Space Pruning in the Flight Planning Problem N2 - We study the Flight Planning Problem for a single aircraft, where we look for a minimum cost path in the airway network, a directed graph. Arc evaluation, such as weather computation, is computationally expensive due to non-linear functions, but required for exactness. We propose several pruning methods to thin out the search space for Dijkstra's algorithm before the query commences. We do so by using innate problem characteristics such as an aircraft's tank capacity, lower and upper bounds on the total costs, and in particular, we present a method to reduce the search space even in the presence of regional crossing costs. We test all pruning methods on real-world instances, and show that incorporating crossing costs into the pruning process can reduce the number of nodes by 90\% in our setting. T3 - ZIB-Report - 20-32 Y1 - 2019 U6 - https://doi.org/https://doi.org/10.4230/OASIcs.ATMOS.2019.8 SN - 1438-0064 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 - TY - CHAP A1 - Gamrath, Gerwin A1 - Reuther, Markus A1 - Schlechte, Thomas A1 - Swarat, Elmar T1 - An LP-based heuristic for Inspector Scheduling T2 - Proceedings of the 13th International Conference on the Practice and Theory of Automated Timetabling - PATAT 2021: Volume I N2 - We present a heuristic based on linear programming (LP) for the integrated tour and crew roster planning of toll enforcement inspectors. Their task is to enforce the proper paying of a distance-based toll on German motorways. This leads to an integrated tour planning and duty rostering problem; it is called Toll Enforcement Problem (TEP). We tackle the TEP by a standard multi-commodity flow model with some extensions in order to incorporate the control tours. The heuristic consists of two variants. The first, called Price & Branch, is a column generation approach to solve the model’s LP relaxation by pricing tour and roster arc variables. Then, we compute an integer feasible solution by restricting to all variables that were priced. The second is a coarse-to-fine approach. Its basic idea is projecting variables to an aggregated variable space, which is much smaller. The aim is to spend as much algorithmic effort in this coarse model as possible. For both heuristic procedures we will show that feasible solutions of high quality can be computed even for large scale industrial instances. Y1 - 2021 UR - https://patatconference.org/patat2020/proceedings/ VL - 1 SP - 77 EP - 86 ER - TY - JOUR A1 - Pronzato, Luc A1 - Sagnol, Guillaume T1 - Removing inessential points in c- and A-optimal design JF - Journal of Statistical Planning and Inference N2 - A design point is inessential when it does not contribute to an optimal design, and can therefore be safely discarded from the design space. We derive three inequalities for the detection of such inessential points in c-optimal design: the first two are direct consequences of the equivalence theorem for c-optimality; the third one is derived from a second-order cone programming formulation of c-optimal design. Elimination rules for A-optimal design are obtained as a byproduct. When implemented within an optimization algorithm, each inequality gives a screening test that may provide a substantial acceleration by reducing the size of the problem online. Several examples are presented with a multiplicative algorithm to illustrate the effectiveness of the approach. Y1 - 2021 UR - https://hal.archives-ouvertes.fr/hal-02868664 U6 - https://doi.org/10.1016/j.jspi.2020.11.011 VL - 213 SP - 233 EP - 252 ER - TY - JOUR A1 - Lindner, Niels T1 - Hypersurfaces with defect JF - Journal of Algebra N2 - A projective hypersurface X⊆P^n has defect if h^i(X) ≠ h^i(P^n) for some i∈{n,…,2n−2} in a suitable cohomology theory. This occurs for example when X⊆P^4 is not Q-factorial. We show that hypersurfaces with defect tend to be very singular: In characteristic 0, we present a lower bound on the Tjurina number, where X is allowed to have arbitrary isolated singularities. For X with mild singularities, we prove a similar result in positive characteristic. As an application, we obtain an estimate on the asymptotic density of hypersurfaces without defect over a finite field. Y1 - 2020 U6 - https://doi.org/10.1016/j.jalgebra.2020.02.022 VL - 555 SP - 1 EP - 35 ER - TY - CHAP A1 - Prause, Felix A1 - Borndörfer, Ralf T1 - A Bayesian Rolling Horizon Approach for Rolling Stock Rotation Planning with Predictive Maintenance T2 - 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024) N2 - We consider the rolling stock rotation planning problem with predictive maintenance (RSRP-PdM), where a timetable given by a set of trips must be operated by a fleet of vehicles. Here, the health states of the vehicles are assumed to be random variables, and their maintenance schedule should be planned based on their predicted failure probabilities. Utilizing the Bayesian update step of the Kalman filter, we develop a rolling horizon approach for RSRP-PdM, in which the predicted health state distributions are updated as new data become available. This approach reduces the uncertainty of the health states and thus improves the decision-making basis for maintenance planning. To solve the instances, we employ a local neighborhood search, which is a modification of a heuristic for RSRP-PdM, and demonstrate its effectiveness. Using this solution algorithm, the presented approach is compared with the results of common maintenance strategies on test instances derived from real-world timetables. The obtained results show the benefits of the rolling horizon approach. Y1 - 2024 U6 - https://doi.org/10.4230/OASIcs.ATMOS.2024.13 VL - 123 SP - 13:1 EP - 13:19 ER - TY - CHAP A1 - Bortoletto, Enrico A1 - Lindner, Niels ED - Voigt, Guido ED - Fliedner, Malte ED - Haase, Knut ED - Brüggemann, Wolfgang ED - Hoberg, Kai ED - Meissner, Jörn T1 - Scaling and Rounding Periodic Event Scheduling Instances to Different Period Times T2 - Operations Research Proceedings 2023 N2 - 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. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-031-58405-3_51 SP - 397 EP - 402 ER - TY - CHAP A1 - Ebert, Patricia A1 - Masing, Berenike A1 - Lindner, Niels A1 - Gleixner, Ambros T1 - Sorting Criteria for Line-based Periodic Timetabling Heuristics T2 - Operations Research Proceedings 2024. OR 2024 N2 - It is well-known that optimal solutions are notoriously hard to find for the Periodic Event Scheduling Problem (PESP), which is the standard mathematical formulation to optimize periodic timetables in public transport. We consider a class of incremental heuristics that have been demonstrated to be effective by Lindner and Liebchen (2023), however, for only one fixed sorting strategy of lines along which a solution is constructed. Thus, in this paper, we examine a variety of sortings based on the number, weight, weighted span, and lower bound of arcs, and test for each setting various combinations of the driving, dwelling, and transfer arcs of lines. Additionally, we assess the impact on the incremental extension of the event-activity network by minimizing resp. maximizing a connectivity measure between subsets of lines. We compare our 27 sortings on the railway instances of the benchmarking library PESPlib within the ConcurrentPESP solver framework. We are able to find five new incumbent solutions, resulting in improvements of up to 2%. Y1 - 2025 SP - 348 EP - 354 ER -