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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.
A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)
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.
We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree property for terminal nodes in hypergraphs which generalizes the well-known graph property that the average degree of terminal nodes in Steiner trees is at most 2.
The Cycle Embedding Problem
(2016)
Given two hypergraphs, representing a fine and a coarse "layer", and a cycle cover of the nodes of the coarse layer, the cycle embedding problem (CEP) asks for an embedding of the coarse cycles into the fine layer. The CEP is NP-hard for general hypergraphs, but it can be solved in polynomial time for graphs. We propose an integer rogramming formulation for the CEP that provides a complete escription of the CEP polytope for the graphical case. The CEP comes up in railway vehicle rotation scheduling. We present computational results for problem instances of DB Fernverkehr AG that justify a sequential coarse-first-fine-second planning approach.
A Simple Way to Compute the Number of Vehicles That Are Required to Operate a Periodic Timetable
(2018)
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.
Dieses Dokument fasst den Stand der mathematischen Modellierung von
Preissystemen des öV mittels eines am ZIB entwickelten Tarifgraphenmodells zusammen. Damit sind sehr einfache und konzise
Beschreibungen von Tarifstrukturen möglich, die sich algorithmisch
behandeln lassen: Durch das zeitgleiche Tracken eines Pfades im
Routinggraphen im Tarifgraphen kann schon während einer Routenberechnung der Preis bestimmt werden. Wir beschreiben
zunächst das Konzept. Die konkrete Realisierung wird im Folgenden
beispielhaft an den Tarifsystemen der Verkehrsverbünde Warnow,
MDV, Vogtland, Bremen/Niedersachsen, Berlin/Brandenburg und Mittelsachsen erläutert. Anschließend folgen Überlegungen zur konkreten Implementierung von Kurzstrecken-Tarifen und zur Behandlung des Verbundübergriffs.
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.
We consider multi-commodity flow problems in which capacities are installed on paths. In this setting, it is often important to distinguish between flows on direct connection routes, using single paths, and flows that include path switching. We show that the concept of metric inequalities to characterize capacities that
support a multi-commodity flow can be generalized to deal with direct connections.