The Effect of Planarization on Width

We study the effects of planarization (the construction of a planar diagram $D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Railway Timetable Optimization

In this cumulative dissertation, we study several aspects of railway timetable optimization. The first contributions cover Practical Applications of Automatic Railway Timetabling. In particular, for the problem of simultaneously scheduling all freight trains in Germany such that there are no conflicts between them, we propose a novel column generation approach. Each train can choose from an iteratively growing set of possible routes and times, so called slots. For the task of choosing maximally many slots without conflicts, we present and apply the heuristic algorithm Conflict Resolving (CR). With these two methods, we are able to schedule more than 5000 trains simultaneously, exceeding the scopes of other studies. A second practical application that we study is measuring the capacity increase in the railway network when equipping freight trains with electro-pneumatic brakes and middle buffer couplings. Methodically, we propose to explicitly construct as many slots as possible for such trains and measure the capacity as the number of constructed slots. Furthermore, we contribute to the field of Algorithms and Computability in Timetable Generation. We present two heuristic solution algorithms for the Maximum Satisfiability Problem (MaxSAT). In the literature, it has been proposed to encode different NP-complete problems that occur in railway timetabling in MaxSAT. In numerical experiments, we prove that our algorithms are competitive to state-of-the-art MaxSAT solvers. Moreover, we study the parameterized complexity status of periodic scheduling and give proofs that the problem is NP-complete for input graphs of bounded treewidth, branchwidth and carvingwidth. Finally, we propose a framework for analyzing Delay Propagation in Railway Networks. More precisely, we develop delay transmission rules based on different correlation measures that can be derived from historical operations data. What is more, we apply SHAP values from Explainable AI to the problem of discerning primary delays that occur stochastically in the operations, to secondary follow-up delays. Transmission rules that are derived from the secondary delays indicate where timetable adjustments are needed. In our last contribution in this field, we apply such adjustment rules for black-box optimization of timetables in a simulation environment