Optimal Algorithms for the Path/Tree-Shaped Facility Location Problems in Trees

Abstract

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).

Minimax regret path location on trees

“…The brackets beside each vertex are the interval weights associated with the vertices. In this simple example we assume that only the weight of vertex v 3 is uncertain and it can vary in the interval [1,12]. The numbers on the edges are the edge lengths.…”

“…Since its publication, several of their results have been improved while they have also contributed to start new interesting research lines (see e.g. [1,12,16,19,25,26,29,[33][34][35]). The aim of this article is to study discrete and continuous minimax regret path location problems on networks with uncertainty on vertex weights.…”

“…. , n. This takes O(n 2 ) time using the algorithm in [12] which computes the weighted path center of a tree in O(n) time. 2.…”

“…Solve the weighted path center on T and find the restriction of this optimal path to T . Using the algorithm by [12], it takes O(n) time.…”