K-delivery Traveling Salesman Problem on Tree Networks

In this paper we study the k-delivery traveling salesman problem (TSP) on trees, a variant of the non-preemptive capacitated vehicle routing problem with pickups and deliveries. We are given n pickup locations and n delivery locations on trees, with exactly one item at each pickup location. The k-delivery TSP is to find a minimum length tour by a vehicle of finite capacity k to pick up and deliver exactly one item to each delivery location. We show that an optimal solution for the k-delivery TSP on paths can be found that allows succinct representations of the routes. By exploring the symmetry inherent in the k-delivery TSP, we design a 3 -approximation algorithm for the k-delivery TSP on trees of arbitrary heights. The ratio can be improved to ( 2− 1 2k ) for the problem on trees of height 2. The developed algorithms are based on the following observation: under certain conditions, it makes sense for a non-empty vehicle to turn around and pick up additional loads. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems