Line Planning and Connectivity

Ralf Borndörfer, Marika Neumann, Marc Pfetsch
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 show in this talk that important results on the Steiner tree
more » ... lytope can be carried over to the SCP case. Furthermore, we generalize the famous relation between undirected and directed Steiner tree formulations. The line planning problem can be described as follows: We are given a public transportation network G = (V, E), a set of (simple) line paths P, and a passenger demand matrix D ∈ IN V ×V , which gives the number of passengers who want to travel between different stations in the network. The edges of G have nonnegative travel times τ ∈ IR E + , the paths have nonnegative costs c ∈ IR P + and capacities κ ∈ IR P +. The problem is to find a set of line paths P ⊆ P with associated frequencies f p ∈ IR + , p ∈ P , and a passenger routing, such that the overall capacities p∈P ,e∈p f p ·κ p on the edges suffice to transport all passengers. There are two possible objectives: to minimize the travel time, or to minimize the cost of the line paths. For a detailed description of the line planning problem see e.g. [1],[2],[3] and the references therein. If we assume large enough capacities the requirement to transport all passengers can usually be replaced by requiring a set of line paths P that connect all stations with positive supply and/or demand. More precisely, let (T, F) be the demand graph of the line planning problem, where T = {v ∈ V | u (d uv +d vu) > 0} is the set of nodes with positive supply or demand, and F = {u, v} | d uv + d vu > 0 a set of demand edges. Then the following holds: If the demand graph is connected, then the set of line paths P of a solution of the line planning problem should also connect all demand nodes. In other words, if we neglect travel times of the passengers, as well as capacities and frequencies of the lines, the line planning problem with connected demand graph reduces to a connectivity Dagstuhl Seminar Proceedings 09261 Models and Algorithms for Optimization in Logistics