Ensemble motion planning in trees

The problem of transporting a set of objects between the vertices of a tree by a unit capacity vehicle that travels along the edges of the tree is considered. For the case in which any object can be dropped at intermediate vertices along its route and picked up later, the authors present algorithms for finding a minimum cost transportation that run in O(k+qn) time and in O(k+n log n) time, where n is the number of vertices in the tree, k is the number of objects to be moved, and q⩽min{k,n} is the number of nontrivial connected components in a related directed graph. In the case in which each object must be carried directly from its initial vertex to its destination, the problem is NP-hard. The authors present approximation algorithms that run in O(k +n) time with a performance ratio of 3/2 and in O(k+n log β(n,q)) time with performance ratio of 5/4