Algorithms for a special class of state-dependent shortest path problems with an application to the train routing problem

We study the state-dependent shortest path problem and focus on its application to the Single Train Routing Problem consisting of a rail network with only double-track segments, where the objective is to route one train through an empty network as fast as possible. We show that the Single Train Routing Problem is NP-hard. We investigate the solution properties and present sufficient conditions for optimality. Different conditions on the parameters are given to guarantee that certain local route selection is optimal. Then, a dynamic programming heuristic is introduced and conditions when the proposed heuristic can obtain the optimal solution in polynomial time are also discussed. Experimental results show the efficiency of the proposed heuristics for general problem settings.