Least-cost path in public transportation systems with fare rebates that are path- and time-dependent

We consider a new class of shortest path problems: path-dependent shortest path, in which the cost of an arc (i, j) in a path P_sj, from some node s to j, is dependent on the arc (i, j) as well as the preceding path P_sj. This class of problems arises when we consider fare rebates in many integrated public transportation systems (buses and subways) where rebates are given to a commuter when he switches service lines. We show that the path-dependent shortest path, in general, is NP-complete whereas its special case, the suffix-k path-dependent shortest path, can be solved by any standard shortest path procedure in polynomial time with a technique called dual path graph transformation. We also discuss a realistic application of path-dependent shortest path in finding least cost path in the context of public transportation system in Singapore where fare rebates are given to commuters when they switch service lines. The fare rebates are path-dependent and time-dependent. We give a fast polynomial time algorithm for this problem that combines past techniques for handling simpler versions of least cost paths problems. We also prove key properties of our model in order to gain computational efficiency.