P-Cable Trench Problem with Covering

We introduce the p-Cable Trench Problem with Covering. Secondary servers are located to provide service within a coverage radius (e.g., Wi-Fi antennas). Each secondary server must be connected to a primary server (a router), through a dedicated cable (wire pair). Trenches need to be dug to bury ducts that contain all the cables. The service must be provided to a whole region, at the minimum cost, consisting of the trench and duct cost and the total cable length cost. The number of primary servers is preset, and they are located on nodes belonging to a set of candidate locations (nodes of an existing telephone network). We propose a linear integer optimization model based on multicommodity flow, which we solve using branch and bound, a linear relaxation and two different Lagrangian relaxation-based heuristics, for which we design ad-hoc procedures to improve the upper bounds. We solve instances of up to 200 nodes, and also an application consisting in locating Wi-Fi antennas in Vina del Mar, Chile.