Recognizing Bundles in Time Table Graphs - A Structural Approach

We are dealing with an application problem arising in a cooperation with the national German railway company. It occurs in the analysis of time table data and involves inferring the underlying railroad network and the actual travel route of the trains when only their time tables are known. The structural basis of our considerations in this paper is a directed graph constructed from train time tables, where train stations correspond to vertices, and where pairs of consecutive stops of trains correspond to edges. Determining the travel route of trains amounts to an edge classification problem in this graph. Exploiting the structure of the graph, we approach the edge classification problem by locating vertices that intuitively correspond to train stations where the underlying railroad network branches into several directions, and that induce a partition of the edge set into bundles. We first describe the modeling process of the classification problem resulting in the bundle recognition problem. Given the NP-hardness of the corresponding optimization problem, we then present a heuristic that makes an educated guess at an initial vertex set of potential bundle end points which is then systematically improved. Finally, we perform a computational study using time table data from 13 European countries.