Routing a vehicle of capacity greater than one

Consider the problem of transporting a set of objects between the vertices of a path by a vehicle which has limited capacity. The problem of finding a shortest tour for the vehicle to transport all objects from their initial vertices to their destination vertices is a fundamental problem in motion planning. It is shown that the problem is NP-complete if every object must be transported directly from its initial vertex to its destination. However, if objects can be dropped at intermediate vertices along its tour and picked up later then the problem can be solved in linear time. It is also shown that if the underlying graph is a tree, instead of a path, then the problem is NP-complete even if objects can be dropped at intermediate vertices.