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DOI: 10.4230/OASIcs.ATMOS.2019.14

An Asymptotically Optimal Approximation Algorithm for the Travelling Car Renter Problem

Authors: Lehilton L. C. Pedrosa, Greis Y. O. Quesquén, and Rafael C. S. Schouery

Published in: OASIcs, Volume 75, 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)

Abstract

In the classical Travelling Salesman Problem (TSP), one wants to find a route that visits a set of n cities, such that the total travelled distance is minimum. An often considered generalization is the Travelling Car Renter Problem (CaRS), in which the route is travelled by renting a set of cars and the cost to travel between two given cities depends on the car that is used. The car renter may choose to swap vehicles at any city, but must pay a fee to return the car to its pickup location. This problem appears in logistics and urban transportation when the vehicles can be provided by multiple companies, such as in the tourism sector. In this paper, we consider the case in which the return fee is some fixed number g >= 0, which we call the Uniform CaRS (UCaRS). We show that, already for this version, there is no o(log n)-approximation algorithm unless P = NP. The main contribution is an O(log n)-approximation algorithm for the problem, which is based on the randomized rounding of an exponentially large LP-relaxation.

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Lehilton L. C. Pedrosa, Greis Y. O. Quesquén, and Rafael C. S. Schouery. An Asymptotically Optimal Approximation Algorithm for the Travelling Car Renter Problem. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019). Open Access Series in Informatics (OASIcs), Volume 75, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{pedrosa_et_al:OASIcs.ATMOS.2019.14,
  author =	{Pedrosa, Lehilton L. C. and Quesqu\'{e}n, Greis Y. O. and Schouery, Rafael C. S.},
  title =	{{An Asymptotically Optimal Approximation Algorithm for the Travelling Car Renter Problem}},
  booktitle =	{19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019)},
  pages =	{14:1--14:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-128-3},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{75},
  editor =	{Cacchiani, Valentina and Marchetti-Spaccamela, Alberto},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2019.14},
  URN =		{urn:nbn:de:0030-drops-114263},
  doi =		{10.4230/OASIcs.ATMOS.2019.14},
  annote =	{Keywords: Approximation Algorithm, Travelling Car Renter Problem, LP-rounding, Separation Problem}
}

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An Asymptotically Optimal Approximation Algorithm for the Travelling Car Renter Problem

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