Abstract: A network is given whose edges need to be constructed (or restored after a disaster). The lengths of edges represent the required construction/restoration times given available resources, and one unit of length of the network can be constructed per unit of time. All points of the network are accessible for construction at any time. For each pair of vertices, a due date is given. It is required to find a construction schedule that minimizes the maximum lateness of all pairs of vertices, where the lateness of a pair is the difference between the time when the pair becomes connected by an already constructed path and the pair's due date. We introduce the problem and analyze its structural properties, present a mixed-integer linear programming formulation, develop a number of lower bounds that are integrated in a branch-and-bound algorithm, and discuss results of computational experiments both for instances based on randomly generated networks and for instances based on 2010 Chilean earthquake data.