The preemptive swapping problem on a tree

This article considers the swapping problem on a tree. In this problem at most one object of some type is available at each vertex, and each vertex also requests at most one object of a given type. The total demand and the total supply of each object type are identical. The problem is to determine a minimum cost routing plan starting and ending at a prespecified vertex which is the depot, for a single vehicle of unit capacity and m object types, so that all vertex requests are satisfied. We consider the preemptive mode in which objects may be temporarily dropped along the way. It is shown that this problem is NP-hard. A heuristic with a worst-case performance ratio of 1.5 is developed. Finally, it is shown that the case where m = 1 is polynomially solvable.