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.

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Keywords (fractional) matching, Integrated railway planning, Treewidth, Vehicle scheduling
Persistent URL dx.doi.org/10.4230/OASIcs.ATMOS.2018.15, hdl.handle.net/1765/110320
Conference 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, ATMOS 2018
van Lieshout, R.N, & Bouman, P.C. (2018). Vehicle scheduling based on a line plan. In OpenAccess Series in Informatics. doi:10.4230/OASIcs.ATMOS.2018.15