Optimal algorithms for the path/tree-shaped facility location problems in trees

Extensive facility location models in networks deal with the location of special types of subgraphs such as paths or trees and can be considered as extensions of classical facility location models. In this paper we consider the problem of locating a path-shaped or tree-shaped (extensive) facility in trees, under the condition that existing facilities are already located. We introduce a parametric-pruning method to solve the conditional discrete/continuous extensive weighted 1-center location problems in trees in linear time. This improves the recent results of O(n log n) by Tamir et al. (J. Algebra 56:50-75, [2005]).