Line Planning and Connectivity

The line planning problem in public transport deals with the construction of a system of lines that is both attractive for the passengers and of low costs for the operator. In general, the computed line system should be connected, i.e., for each two stations there have to be a path that is covered by the lines. This subproblem is a generalization of the well-known Steiner tree problem; we call it the Steiner connectivity Problem. We discuss complexity of this problem, generalize the so-called Steiner partition inequalities and give a transformation to the directed Steiner tree problem. We show that directed models provide tight formulations for the Steiner connectivity problem, similar as for the Steiner tree problem