On the Linking Set Problem

We consider the Linking Set Problem which occurs as a subproblem in a decomposition procedure to solve large scale p-median problems such as the Optimal Diversity Management Problem. We show that a greedy algorithm solves the Linking Set Problem when the costs have a special structure. This special structure usually occurs when the Linking Set Problem arises from the decomposition of location problems as a consequence of a well-known property of the objective functions of the location problems. We also provide a combinatorial algorithm to solve the linear relaxation problem based on the lagrangean relaxation and show that there is no linear gap in the case the particular structure of the costs is verified. For general costs we compare two exact algorithm that have been proposed in the literature.