Approximation algorithms for a vehicle routing problem

In this paper we investigate a vehicle routing problem motivated by a real-world application in cooperation with the German Automobile Association ( ADAC). The general task is to assign service requests to service units and to plan tours for the units such as to minimize the overall cost. The characteristics of this large-scale problem due to the data volume involve strict real-time requirements. We show that the problem of finding a feasible dispatch for service units starting at their current position and serving at most k requests is NP-complete for each fixed k >= 2. We also present a polynomial time (2k - 1)- approximation algorithm, where again k denotes the maximal number of requests served by a single service unit. For the boundary case when k equals the total number vertical bar E vertical bar of requests (and thus there are no limitations on the tour length), we provide a (2 - 1/vertical bar E vertical bar)-approximation. Finally, we extend our approximation results to include linear and quadratic lateness costs.