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Vehicle Scheduling Based on a Line Plan

Authors Rolf N. van Lieshout, Paul C. Bouman

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Rolf N. van Lieshout
  • Erasmus University Rotterdam, Econometric Institute, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands
Paul C. Bouman
  • Erasmus University Rotterdam, Econometric Institute, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands

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Rolf N. van Lieshout and Paul C. Bouman. Vehicle Scheduling Based on a Line Plan. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018). Open Access Series in Informatics (OASIcs), Volume 65, pp. 15:1-15:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


We consider the following problem: given a set of lines in a public transportation network with their round trip times and frequencies, a maximum number of vehicles and a maximum number of lines that can be combined into a vehicle circulation, does there exist a set of vehicle circulations that covers all lines given the constraints. Solving this problem provides an estimate of the costs of operating a certain line plan, without having to compute a timetable first. We show that this problem is NP-hard for any restriction on the number of lines that can be combined into a circulation which is equal to or greater than three. We pay special attention to the case where at most two lines can be combined into a circulation, which is NP-hard if a single line can be covered by multiple circulations. If this is not allowed, a matching algorithm can be used to find the optimal solutions, which we show to be a 16/15-approximation for the case where it is allowed. We also provide an exact algorithm that is able to exploit low tree-width of the so-called circulation graph and small numbers of vehicles required to cover single circulations.

Subject Classification

ACM Subject Classification
  • Applied computing → Transportation
  • Mathematics of computing → Graph algorithms
  • Vehicle scheduling
  • integrated railway planning
  • (fractional) matching
  • treewidth


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