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Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey results on cycle bases and prove new ones. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results about them, in particular, a-priori length bounds. We give polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

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A short time ago, decision support by operations research methods in railway companies was limited to operations planning (e.g., vehicle scheduling, duty scheduling, crew rostering). In effect since December 12, 2004, the 2005 timetable of the Berlin subway is based on the results of mathematical programming techniques. It is the first such service concept that has been put into daily operation. Profiting from these techniques, compared with the previous timetable, the Berlin subway today operates with a timetable that offers shorter passenger waiting times—both at stops and at transfers—and even saves one train. The work is based on a well-established graph model, the periodic event-scheduling problem (Pesp). This model was introduced as early as 1989. Besides describing in detail its first success story in practice, in this paper we also deepen a result on the asymptotic complexity of the Pesp: we provide MAXSNP-hardness proofs of two natural optimization variants.

Abstract. In the planning process of railway companies, we propose to integrate important decisions of network planning, line planning, and vehicle scheduling into the task of periodic timetabling. From such an integration, we expect to achieve an additional potential for optimization.Models for periodic timetabling are commonly based on the Periodic Event Scheduling Problem (PESP). We show that, for our purpose of this integration, the PESP has to be extended by only two features, namely a linear objective function and a symmetry requirement. These extensions of the PESP do not really impose new types of constraints. Indeed, practitioners have already required them even when only planning timetables autonomously without interaction with other planning steps. Even more important, we only suggest extensions that can be formulated by mixed integer linear programs.Moreover, in a selfcontained presentation we summarize the traditional PESP modeling capabilities for railway timetabling. For the first time, also special practical requirements are considered that we proove not being expressible in terms of the PESP.