A Graph- and Monoid-Based Framework for Price-Sensitive Routing in Local Public Transportation Networks

Authors Ricardo Euler , Ralf Borndörfer

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Author Details

Ricardo Euler
  • Zuse Institute Berlin, Germany
Ralf Borndörfer
  • Zuse Institute Berlin, Germany


We thank Niels Lindner and Pedro Maristany de las Casas for many fruitful discussions on the subject as well as MDV and InfraDialog GmbH for providing the data for this study.

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Ricardo Euler and Ralf Borndörfer. A Graph- and Monoid-Based Framework for Price-Sensitive Routing in Local Public Transportation Networks. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We present a novel framework to mathematically describe the fare systems of local public transit companies. The model allows the computation of a provably cheapest itinerary even if prices depend on a number of parameters and non-linear conditions. Our approach is based on a ticket graph model to represent tickets and their relation to each other. Transitions between tickets are modeled via transition functions over partially ordered monoids and a set of symbols representing special properties of fares (e.g. surcharges). Shortest path algorithms rely on the subpath optimality property. This property is usually lost when dealing with complicated fare systems. We restore it 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. By integrating our framework in the multi-criteria RAPTOR algorithm we provide a price-sensitive algorithm for the earliest arrival problem and assess its performance on data obtained from MDV. We discuss three preprocessing techniques that improve run times enough to make the algorithm applicable for real-time queries.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Applied computing → Transportation
  • shortest path
  • public transit
  • optimization
  • price-sensitive
  • raptor
  • fare
  • operations research


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