We tackle an optimization problem arising in the design of sensor networks: given a set of sensors, only one being connected to a backbone, to establish connection routes from each of them to the sink. Under a shortest path routing protocol, the set of connections form a spanning tree. Energy is required to transmit and receive data, and sensors have limited battery capacity: as soon as one sensor runs out of battery, a portion of the network is disconnected. We, therefore, search for the spanning tree maximizing the time elapsed before such a disconnection occurs, and therefore, maintenance is required. We propose new mathematical formulations for the problem, proving and exploiting theoretical results on its combinatorial structure. On that basis, we design algorithms offering a priori guarantees of global optimality. We undertake an extensive experimental campaign, showing our algorithms to outperform previous ones from the literature by orders of magnitude. We also identify which instance features have higher impact on network lifetime.

This paper deals with the multiple vehicle balancing problem (MVBP). Given a fleet of vehicles of limited capacity, a set of vertices with initial and target inventory levels and a distribution network, the MVBP requires to design a set of routes along with pickup and delivery operations such that inventory is redistributed among the vertices without exceeding capacities, and routing costs are minimized. The MVBP is NP‐hard, generalizing several problems in transportation, and arising in bike‐sharing systems. Using theoretical properties of the problem, we propose an integer linear programming formulation and introduce strengthening valid inequalities. Lower bounds are computed by column generation embedding an ad‐hoc pricing algorithm, while upper bounds are obtained by a memetic algorithm that separate routing from pickup and delivery operations. We combine these bounding routines in both exact and matheuristic algorithms, obtaining proven optimal solutions for MVBP instances with up to 25 stations.