Hop Constrained Connected Facility Location: A Hierarchy of Formulations

Connected Facility Location (ConFL) is a problem that combines network design and facility location aspects: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized. In the Hop Constrained Facility Location Problem (HC ConFL) the number of edges between the root and any open facility should not exceed a given number H > 1. We develop 16 mixed integer programming models for this problem. In branch-and-bound frameworks the quality of linear programming lower bounds of these formulations is of particular interest. In this theoretical study we compare the relative quality of these relaxations and provide a hierarchy of the corresponding models. This paper comprises a first theoretical study on polyhedral aspects of this problem of great practical importance in the design of telecommunication or data-management networks.