Nonpreemptive Ensemble Motion Planning on a Tree

Abstract

Consider the problem of finding a minimum-cost tour to transport a set of objects between the vertices of an edge-weighted 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 destination then finding such a minimum-cost tour is NP-hard. Several fast approximation algorithms are presented for this problem. The fastest runs in O( k + n) time and generates a tour of cost at most 3 2 times the cost of a minimum-cost tour, where n is the number of vertices in the tree, k is the number of objects to be moved. Another runs in O( k + n log β( n, q)) time and generates a tour of cost at most 5 4 times the cost of a minimum-cost tour, where q ≤ min{ k, n} is the number of nontrivial connected components in a related directed graph, and β( n, q) = min{ i | log i n ≤ n/ q}. In addition, a family of polynomial-time approximation algorithms is given, whose ratio of the cost of tour generated to the cost of a minimum-cost tour decreases to at most 1.21363 as the running time increases.