Vehicle routing in simple graphs

Consider the problem of finding an optimal route for transporting a set of objects between the vertices of a graph by a vehicle that travels along the edges of the graph. We focus on simple graphs such as paths, trees and cycles. We consider various versions of the problem, based on the capacity of the vehicle and whether or not objects can be dropped at intermediate vertices. For the case in which the vehicle has unit capacity, we focus on trees. Let n be the number of vertices, k be the number of objects to be transported, and $q \leq$ min$\{ k,n\}$ be the number of components in the balanced graph. We show that the problem can be solved in $O(k + qn)$ time if objects can be dropped at intermediate vertices. However, we show that finding a shortest route is NP-hard if every object must be carried directly from its initial vertex to its destination. We present several approximation algorithms for this version of the problem. The fastest one runs in $O(k + n)$ time with a performance ratio of 3/2. The approximation algorithm with the best performance ratio runs in $O(k + n\log\beta(n,q))$ time, and generates a transportation of cost at most 5/4 times the cost of an optimal transportation, where $\beta(n,q)$ is a very slowly growing function. For the case in which the vehicle has capacity greater than one, we focus on paths and trees. We show that if the graph is a path and objects can be dropped at intermediate vertices, then the problem can be solved in $O(k + n)$ time. However, we show that the problem is NP-hard if every object must be carried directly from its initial vertex to its destination. In the case that the graph is a tree, we show that the problem is NP-hard even if objects can be dropped at the intermediate vertices. The version of the problem in which the vehicle has infinite capacity is also known as the courier problem. We show that the courier problem on paths can be solved in $O(k + n)$ time, and the courier problem on cycles can be solved in $O(k + n\sp2)$ time. We also show that the courier problem in trees is NP-hard