The 2-radius and 2-radiian problems on trees

In this paper, we consider two facility location problems on tree networks. One is the 2-radius problem, whose goal is to partition the vertex set of the given network into two non-empty subsets such that the sum of the radii of these two induced subgraphs is minimum. The other is the 2-radiian problem, whose goal is to partition the network into two non-empty subsets such that the sum of the centdian values of these two induced subgraphs is minimum. We propose an O(n)-time algorithm for the 2-radius problem on trees and an O(nlogn)-time algorithm for the 2-radiian problem on trees, where n is the number of vertices in the given tree.