Location of paths on trees with minimal eccentricity and superior section

Network location theory has traditionally been concerned with the optimal location of a single-point facility at either a vertex or along an arc in the network. Recently, some authors have departed from this traditional problem and have considered the location of extensive facilities, such as paths, trees or cycles. In this paper, we consider the optimal location of paths on trees with regard to two objective functions: the eccentricity and the superior section. We first present methods to find paths with minimal eccentricity and minimal superior section on trees with arbitrary positive lengths. Then, we analyse the biobjective optimization problem and propose an algorithm, based on a progressive reduction of the initial tree, to obtain all efficient paths. Modifications of the proposed algorithm to solve the problem when a general objective function is used instead of the eccentricity function are also given.