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# An Asymptotically Optimal Approximation Algorithm for the Travelling Car Renter Problem

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## Cite As

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)
https://doi.org/10.4230/OASIcs.ATMOS.2019.14

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
• Theory of computation → Routing and network design problems
##### Keywords
• Approximation Algorithm
• Travelling Car Renter Problem
• LP-rounding
• Separation Problem

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