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• English  (260)
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• 1
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Publication Date: 2020-03-09
Description: In this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so-called {\em bordered block diagonal form}. More precisely, given some matrix $A$, we try to assign as many rows as possible to some number of blocks of limited size such that no two rows assigned to different blocks intersect in a common column. Bordered block diagonal form is desirable because it can guide and speed up the solution process for linear and integer programming problems. We show that various matrices from the %LP- and MIP-libraries \Netlib{} and MIPLIB can indeed be decomposed into this form by computing optimal decompositions or decompositions with proven quality. These computations are done with a branch-and-cut algorithm based on polyhedral investigations of the matrix decomposition problem.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 2
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Publication Date: 2020-08-05
Description: This paper is about {\em set packing relaxations\/} of combinatorial optimization problems associated with acyclic digraphs and linear orderings, cuts and multicuts, and vertex packings themselves. Families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 3
Publication Date: 2020-08-05
Description: This article investigates a certain class of combinatorial packing problems and some polyhedral relations between such problems and the set packing problem.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 4
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Publication Date: 2020-08-05
Description: We propose an efficient column generation method to minimize the probability of delay propagations along aircraft rotations. In this way, delay resistant schedules can be constructed. Computational results for large-scale real-world problems demonstrate substantial punctuality improvements. The method can be generalized to crew and integrated scheduling problems.
Keywords: ddc:510
Language: English
Type: reportzib , doc-type:preprint
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• 5
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Publication Date: 2020-08-05
Description: We propose a novel integer programming approach to transfer minimization for line planning problems in public transit. The idea is to incorporate penalties for transfers that are induced by “connection capacities” into the construction of the passenger paths. We show that such penalties can be dealt with by a combination of shortest and constrained shortest path algorithms such that the pricing problem for passenger paths can be solved efficiently. Connection capacity penalties (under)estimate the true transfer times. This error is, however, not a problem in practice. We show in a computational comparison with two standard models on a real-world scenario that our approach can be used to minimize passenger travel and transfer times for large-scale line planning problems with accurate results.
Keywords: ddc:510
Language: English
Type: reportzib , doc-type:preprint
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• 6
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Publication Date: 2020-12-15
Description: This paper introduces the "line connectivity problem", a generalization of the Steiner tree problem and a special case of the line planning problem. We study its complexity and give an IP formulation in terms of an exponential number of constraints associated with "line cut constraints". These inequalities can be separated in polynomial time. We also generalize the Steiner partition inequalities.
Keywords: ddc:510
Language: English
Type: reportzib , doc-type:preprint
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• 7
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Publication Date: 2020-08-05
Description: We consider an auction of slots to run trains through a railway network. In contrast to the classical setting for combinatorial auctions, there is not only competition for slots, but slots can mutually exclude each other, such that general conflict constraints on bids arise. This turns the winner determination problem associated with such an auction into a complex combinatorial optimization problem. It also raises a number of auction design questions, in particular, on incentive compatibilty. We propose a single-shot second price auction for railway slots, the Vickrey Track Auction (VTA). We show that this auction is incentive compatible, i.e., rational bidders are always motivated to bid their true valuation, and that it produces efficient allocations, even in the presence of constraints on allocations. These properties are, however, lost when rules on the submission of bids such as, e.g., lowest bids, are imposed. Our results carry over to generalized" Vickrey auctions with combinatorial constraints.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 8
Publication Date: 2020-08-05
Description: Technical restrictions and challenging details let railway traffic become one of the most complex transportation systems. Routing trains in a conflict-free way through a track network is one of the basic scheduling problems for any railway company. This article focuses on a robust extension of this problem, also known as train timetabling problem (TTP), which consists in finding a schedule, a conflict free set of train routes, of maximum value for a given railway network. However, timetables are not only required to be profitable. Railway companies are also interested in reliable and robust solutions. Intuitively, we expect a more robust track allocation to be one where disruptions arising from delays are less likely to be propagated causing delays of subsequent trains. This trade-off between an efficient use of railway infrastructure and the prospects of recovery leads us to a bi-criteria optimization approach. On the one hand we want to maximize the profit of a schedule, that is more or less to maximize the number of feasible routed trains. On the other hand if two trains are scheduled as tight as possible after each other it is clear that a delay of the first one always affects the subsequent train. We present extensions of the integer programming formulation in [BorndoerferSchlechte2007] for solving (TTP). These models can incorporate both aspects, because of the additional track configuration variables. We discuss how these variables can directly be used to measure a certain type of robustness of a timetable. For these models which can be solved by column generation techniques, we propose so-called scalarization techniques, see [Ehrgott2005], to determine efficient solutions. Here, an efficient solution is one which does not allow any improvement in profit and robustness at the same time. We prove that the LP-relaxation of the (TTP) including an additional $\epsilon$-constraint remains solvable in polynomial time. Finally, we present some preliminary results on macroscopic real-world data of a part of the German long distance railway network.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 9
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Publication Date: 2020-12-15
Description: The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem, namely, the Steiner cut and the Steiner partition inequalities, as well as the associated polynomial time separation algorithms, can be generalized. Similar to the Steiner tree case, a directed formulation, which is stronger than the natural undirected one, plays a central role.
Keywords: ddc:000
Language: English
Type: reportzib , doc-type:preprint
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• 10
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Publication Date: 2020-08-05
Description: Every day, millions of people are transported by buses, trains, and airplanes in Germany. Public transit (PT) is of major importance for the quality of life of individuals as well as the productivity of entire regions. Quality and efficiency of PT systems depend on the political framework (state-run, market oriented) and the suitability of the infrastructure (railway tracks, airport locations), the existing level of service (timetable, flight schedule), the use of adequate technologies (information, control, and booking systems), and the best possible deployment of equipment and resources (energy, vehicles, crews). The decision, planning, and optimization problems arising in this context are often gigantic and “scream” for mathematical support because of their complexity. This article sketches the state and the relevance of mathematics in planning and operating public transit, describes today’s challenges, and suggests a number of innovative actions. The current contribution of mathematics to public transit is — depending on the transportation mode — of varying depth. Air traffic is already well supported by mathematics. Bus traffic made significant advances in recent years, while rail traffic still bears significant opportunities for improvements. In all areas of public transit, the existing potentials are far from being exhausted. For some PT problems, such as vehicle and crew scheduling in bus and air traffic, excellent mathematical tools are not only available, but used in many places. In other areas, such as rolling stock rostering in rail traffic, the performance of the existing mathematical algorithms is not yet sufficient. Some topics are essentially untouched from a mathematical point of view; e.g., there are (except for air traffic) no network design or fare planning models of practical relevance. PT infrastructure construction is essentially devoid of mathematics, even though enormous capital investments are made in this area. These problems lead to questions that can only be tackled by engineers, economists, politicians, and mathematicians in a joint effort. Among other things, the authors propose to investigate two specific topics, which can be addressed at short notice, are of fundamental importance not only for the area of traffic planning, should lead to a significant improvement in the collaboration of all involved parties, and, if successful, will be of real value for companies and customers: • discrete optimal control: real-time re-planning of traffic systems in case of disruptions, • model integration: service design in bus and rail traffic. Work on these topics in interdisciplinary research projects could be funded by the German ministry of research and education (BMBF), the German ministry of economics (BMWi), or the German science foundation (DFG).
Keywords: ddc:510
Language: English
Type: reportzib , doc-type:preprint
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