The continuous stop location problem in public transportation networks

In this paper we consider the location of stops along the edges of an already existing public transportation network. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network.The positive effect of new stops is given by the better access of the potential customers to public transportation, while the travel time increases due to the additional stopping activities of the trains. The latter leads to a negative effect for the customers. The goal is to cover all given demand points with a minimal amount of additional traveling time, where covering may be defined with respect to an arbitrary norm or gauge. This problem is NP-hard, even in the special case of Euclidean distances. In this paper, we derive a finite candidate set leading to a discrete set covering problem. Moreover, we identify network structures in which the coefficient matrix of the resulting set covering problem is totally unimodular, such that the problem can be solved efficiently in this case. Extensions of the problem are also discussed.