Price Optimal Routing in Public Transportation
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With the development of fast routing algorithms for public transit the optimization of criteria other than arrival time has moved into the spotlight. Due to the often intricate nature of fare regulations, the price of a journey, while certainly being one of the most interesting criteria, is seldom considered. In this work, we present a novel framework to mathematically describe the fare systems of local public transit companies. The model allows us to solve the price-sensitive earliest arrival problem (PSEAP) even if prices depend on a number of parameters and non-linear conditions. We base our approach on conditional fare networks (CFN). A CFN consists of a ticket graph modeling tickets and the relations between them and a ticket transition function defined over a partially ordered monoid and a set of fare events. Typically, shortest path algorithms rely on the subpath optimality property which is usually lost when dealing with intricate fare systems. We show how subpath optimality can be restored by relaxing domination rules for tickets depending on the structure of the ticket graph. An exemplary model for the fare system of Mitteldeutsche Verkehrsbetriebe (MDV) is provided. PSEAP is NP-hard, even when considering a single-criterion version that optimizes for price only. Instances on finite monoids are, however, polynomial time solvable when the monoid is considered part of the encoding length. By integrating our framework in the multi-criteria RAPTOR algorithm, we provide an algorithm for PSEAP and assess its performance on data obtained from MDV. We introduce two simple speed-up techniques that in combination with the recently introduced Tight-BMRAPTOR pruning scheme allow us to solve PSEAP in mere milliseconds on our data set.
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