Approximation schemes for Euclidean vehicle routing problems with time windows

The vehicle routing problem with time windows (VRPTW) is a variant of the classical vehicle routing problem. The paper considers two dimensional and one dimensional VRPTW, in which each demand must be serviced within the time window which is designated by its customer. In the two dimensional problem, each customer has the same unit demand. The paper gives a quasi-polynomial time approximation scheme and an asymptotic polynomial time approximation scheme for the two dimensional and one dimensional problems under the Euclidean setting, respectively. With reasonable vehicle speed requirements, our algorithms could generate the solutions whose the total route length is (1 + O(ε)) times of that of the optimum solutions.