90-02 Research exposition (monographs, survey articles)
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We present a new label-setting algorithm for the Multiobjective Shortest Path (MOSP) problem that computes the minimal complete set of efficient paths for a given instance. The size of the priority queue used in the algorithm is bounded by the number of nodes in the input graph and extracted labels are guaranteed to be efficient. These properties allow us to give a tight output-sensitive running time bound for the new algorithm that can almost be expressed in terms of the running time of Dijkstra's algorithm for the Shortest Path problem. Hence, we suggest to call the algorithm \emph{Multiobjective Dijkstra Algorithm} (MDA). The simplified label management in the MDA allows us to parallelize some subroutines. In our computational experiments, we compare the MDA and the classical label-setting MOSP algorithm by Martins', which we improved using new data structures and pruning techniques. On average, the MDA is $\times2$ to $\times9$ times faster on all used graph types. On some instances the speedup reaches an order of magnitude.
Mathematik im Verkehr
(2014)
Nach einem kurzen Abriss über die Bedeutung des Verkehrssektors als eine wichtige Schlüsseltechnologie im gesamten Verlauf der Menschheitsgeschichte skizzieren wir die Rolle der Mathematik für Verkehr und Transport. Wir spekulieren dann über zukünftige Entwicklungen, insbesondere im Bereich des öffentlichen Personenverkehrs, und begründen, dass die in diesem Bereich anstehenden Herausforderungen nur mit dem Einsatz mathematischer Methoden angemessen bewältigt werden können. Die demographischen Prozesse, die in verschiedenen Teilen der Welt unterschiedlich verlaufen, wie z.B. Überalterung in Europa oder dynamische Trends zu Megastädten in Entwicklungsländern, sich ändernde Lebens- und Produktionsverhältnisse, stark wachsender Bedarf nach Mobilität und enormes Anwachsen der Komplexität der Verkehrsplanung und -durchführung, machen einen verstärkten Zugriff auf mathematische Modellierung, Simulation und Optimierung notwendig. Diese Entwicklung stellt sowohl große Herausforderungen an die Mathematik, wo vielfach noch keine geeigneten Methoden vorhanden sind, als auch an die Praktiker im Bereich von Verkehr und Transport, die sich mit neuen Planungs- und Steuerungstechnologien befassen und diese effizient einsetzen müssen. Hier wird intensive Kooperation zwischen vielen beteiligten Akteuren gefragt sein.
This cumulative thesis collects the following six papers for obtaining the
habilitation at the Technische Universität Berlin, Fakultät II – Mathematik
und Naturwissenschaften:
(1) Set packing relaxations of some integer programs.
(2) Combinatorial packing problems.
(3) Decomposing matrices into blocks.
(4) A bundle method for integrated multi-depot vehicle and duty scheduling
in public transit.
(5) Models for railway track allocation.
(6) A column-generation approach to line planning in public transport.
Some changes were made to the papers compared to the published versions.
These pertain to layout unifications, i.e., common numbering, figure, table,
and chapter head layout. There were no changes with respect to notation or
symbols, but some typos have been eliminated, references updated, and some
links and an index was added. The mathematical content is identical.
The papers are about the optimization of public transportation systems,
i.e.,
bus networks, railways, and airlines, and its mathematical foundations,
i.e.,
the theory of packing problems. The papers discuss mathematical models,
theoretical analyses, algorithmic approaches, and computational aspects of
and to problems in this area.
Papers 1, 2, and 3 are theoretical. They aim at establishing a theory of
packing problems as a general framework that can be used to study traffic
optimization problems. Indeed, traffic optimization problems can often be
modelled as path packing, partitioning, or covering problems, which lead
directly to set packing, partitioning, and covering models. Such models are
used in papers 4, 5, and 6 to study a variety of problems concerning the
planning
of line systems, buses, trains, and crews. The common aim is always
to exploit as many degrees of freedom as possible, both at the level of the
individual problems by using large-scale integer programming techniques, as
well as on a higher level by integrating hitherto separate steps in the
planning
process.