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.

The Traveling Car Renter Problem (CaRS) is a generalization of the Traveling Salesman Problem. This paper presents a hybrid metaheuristic approach to deal with CaRS: an evolutionary algorithm (ScA) and the hybrid method Adaptive Local Search Procedure (ALSP), denoted by ScA+ALSP. A mixed integer programming model proposed for CaRS is corrected and used within the ALSP. The results of experimental studies using a suite of 21 instances taken from the literature indicated that the hybrid ScA+ALSP is competitive regarding the best known algorithm in literature for non-Euclidean CaRS instances. Three new best results are reported.