The Cheapest Ticket Problem in Public Transport

Route choice models in public transport have been discussed for a long time. The main factor why a passenger chooses a specific path is usually based on its length or travel time. However, also the ticket price that passengers have to pay may influence their decision since passengers prefer cheaper paths over more expensive ones. In this paper, we deal with the cheapest ticket problem which asks for a cheapest ticket to travel between a pair of stations. The complexity and the algorithmic approach to solve this problem depend crucially on the underlying fare structure, e.g., it is easy if the ticket prices are proportional to the distance traveled (as in distance tariff fare structures), but may become NP-complete in zone tariff fare structures. We hence discuss the cheapest ticket problem for different variations of distance- and zone-based fare structures. We start by modeling the respective fare structure mathematically, identify its main properties, and finally provide a polynomial algorithm, or prove NP-completeness of the cheapest ticket problem. We also provide general results on the combination of two fare structures, which is often observed in practice.