The 2-facility centdian network problem

The p-facility centdian network problem consists of finding the p points that minimize a convex combination of the p-center and p-median objective functions. The vertices and local centers constitute a dominating set for the 1-facility centdian; i.e., it contains an optimal solution for all instances of the problem. Hooker et al. (1991) give a theoretical result to extend the dominating sets for the 1-facility problems to the corresponding p-facility problems. They claim that the set of vertices and local centers is also a dominating set for the p-facility centdian problem. We give a counterexample and an alternative finite dominating set for p=2. We propose a solution procedure for a network that improves the complexity of the exhaustive search in the dominating set. We also provide a very efficient algorithm that solves the 2-centdian on a tree network with complexity O(n²).