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Consider the problem of transporting a set of objects between the vertices of a tree by a vehicle that travels along the edges of the tree. The vehicle can carry only one object at a time, and it starts and finishes at the same vertex of the tree. It is shown that if each object must be carried directly from its initial vertex to its destina.tion then finding a minimum cost transportation is NPhard. Several fast approximation algorithms are presented for this problem. The fastest runs in O(k + n) time and generates a transportation of cost at most 3/2 times the cost of an optimal transportation, where n is the number of vertices in the tree, k is the number or objects to be moved. Another runs in O(k + nlogp'(n,q)) time, and generates a transportation of cost at most 5/4 times the cost of an optimal transportation, where q~min{k,n} is the number of nontrivial connected components in a related directed graph.

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Consider the problem of finding a minimum cost tour to transport a set of objects uetween the vertices of a tree by a vehicle that travels along the edges of the tree. The vehicle can carry only one object at a time, and it starts and finishes at the same vertex of the tree. It is shown that if objects can ue dropped at intermediate vertices along its route and picked up later, then the prol>lem can be solved in polynomial time. Two efficient algorithms are presellted fol' this problC!m. The first algorithm runs in O(k +qn) time, where n is the number of vertices in the tree, k 1s the number of objects to be moved, and q 5 min {k, n} is the number of nontrivial connected components in a related directed graph. The second algorithm runs in O(k +n log n) time.