## Fastest, average and quantile schedule

Please always quote using this URN: urn:nbn:de:0297-zib-53592

- We consider problems concerning the scheduling of a set of trains on a single track. For every pair of trains there is a minimum headway, which every train must wait before it enters the track after another train. The speed of each train is also given. Hence for every schedule - a sequence of trains - we may compute the time that is at least needed for all trains to travel along the track in the given order. We give the solution to three problems: the fastest schedule, the average schedule, and the problem of quantile schedules. The last problem is a question about the smallest upper bound on the time of a given fraction of all possible schedules. We show how these problems are related to the travelling salesman problem. We prove NP-completeness of the fastest schedule problem, NP-hardness of quantile of schedules problem, and polynomiality of the average schedule problem. We also describe some algorithms for all three problems. In the solution of the quantile problem we give an algorithm, based on a reverse search method, generating with polynomial delay all Eulerian multigraphs with the given degree sequence and a bound on the number of such multigraphs. A better bound is left as an open question.

Author: | Armin Fügenschuh, Konstanty Junosza-Szaniawski, Torsten Klug, Slawomir Kwasiborski, Thomas Schlechte |
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Document Type: | ZIB-Report |

Tag: | eulerian multigraphs; scheduling |

MSC-Classification: | 05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Axx Enumerative combinatorics For enumeration in graph theory, see 05C30 |

CCS-Classification: | G. Mathematics of Computing / G.2 DISCRETE MATHEMATICS / G.2.1 Combinatorics (F.2.2) / Permutations and combinations |

Date of first Publication: | 2015/10/02 |

Series (Serial Number): | ZIB-Report (14-49) |

ISSN: | 1438-0064 |

Published in: | Appeared in: SOFSEM 2015: Theory and Practice of Computer Science |

Licence (German): | Creative Commons - Namensnennung-Nicht kommerziell |