## Abstract

Dial-a-Ride problems consist of a set V of n vertices in a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs {(si,ti)}mi=1, where each object requires to be moved from its source to destination vertex. In the multi-vehicle Dial-a-Ride problem, there are q vehicles each having capacity k and where each vehicle j ∈ [q] has its own depot-vertex rj ∈ V. A feasible schedule consists of a capacitated route for each vehicle (where vehicle j originates and ends at its depot rj) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e. makespan) of vehicles, where the completion time of vehicle j is the time when it returns to its depot rj at the end of its route. We consider the preemptive version of multi-vehicle Dial-a-Ride, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Approximation algorithms for the single vehicle Dial-a-Ride problem (q = 1) have been considered in [3,10]. Our main results are an O(log3n)-approximation algorithm for preemptive multi-vehicle Dial-a-Ride, and an improved O(logt)-approximation for its special case when there is no capacity constraint (here t ≤ n is the number of distinct depot-vertices). There is an Ω(log1/4n) hardness of approximation known [9] even for single vehicle capacitated preemptive Dial-a-Ride.

We also obtain an improved constant factor approximation algorithm for the uncapacitated multi-vehicle problem on metrics induced by graphs excluding any fixed minor (eg. planar metrics). In this case, we obtain improved guarantees of O(log2 n) for capacitated multi-vehicle Dial-a-Ride, and O(1) for the uncapacitated problem.

We also obtain an improved constant factor approximation algorithm for the uncapacitated multi-vehicle problem on metrics induced by graphs excluding any fixed minor (eg. planar metrics). In this case, we obtain improved guarantees of O(log2 n) for capacitated multi-vehicle Dial-a-Ride, and O(1) for the uncapacitated problem.

Original language | English |
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Title of host publication | Algorithms - ESA 2009 |

Publisher | Springer |

Publication date | 2009 |

Pages | 540-552 |

ISBN (Print) | 978-3-642-04127-3 |

DOIs | |

Publication status | Published - 2009 |

Event | 17th European Symposium on Algorithms - København, Denmark Duration: 7 Sept 2009 → 9 Sept 2009 Conference number: 17 http://algo2009.itu.dk/esa-2009 |

### Conference

Conference | 17th European Symposium on Algorithms |
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Number | 17 |

Country/Territory | Denmark |

City | København |

Period | 07/09/2009 → 09/09/2009 |

Internet address |

Series | Lecture Notes in Computer Science |
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Number | Volume 5757/2009 |

ISSN | 0302-9743 |