In this paper, the quadratic p-median optimization problem is discussed, where the goal is to connect users to a selected group of facilities (emergency services, telecommunications servers, healthcare facilities) at the lowest possible cost. The problem is aimed at minimizing the cost of connecting these selected facilities. The costs are symmetric, meaning connecting two different points is the same in both directions. This problem extends the traditional p-median problem, a combinatorial problem used in various fields like facility location, network design, transportation, supply chain networks, emergency services, healthcare, and education planning. Surprisingly, the quadratic version has not been thoroughly considered in the literature. The paper highlights the formulation of two mixed-integer quadratic programming models to find optimal solutions to this problem. One model is a classic formulation, and the other is based on set cover theory. Linear versions and Bender’s decomposition formulations for each model are also derived. A Bender’s decomposition is solved using an algorithm that adds constraints during each iteration to improve the solution. Lazy constraints in the Gurobi solver’s branch and cut algorithm are dynamically added whenever a mixed-integer programming solution is found. Additionally, an efficient local search meta-heuristic is proposed that usually finds optimal solutions for tested instances. Challenging instances with up to 60 facilities and 2000 users are successfully solved. Our results show that Bender’s models with lazy constraints are the most effective for Euclidean and random test cases, achieving optimal solutions in less CPU time. The meta-heuristic also finds near-optimal solutions rapidly for these cases.

In this paper, we analyze the problem of managing users from different slices connecting to a software‐defined network (SDN). We seek to minimize the propagation latency between switches and controllers as well as between controllers themselves. We also minimize the connection latency between users and their network access nodes. Thus, the main highlights of the paper are to formally represent the problem utilizing two equivalent mixed‐integer quadratic programming models. The first one represents the user requirements of each slice by using a membership matrix. The second one consists of subsets of users separated within each slice requirement. Subsequently, the above models are analyzed in a standard linearized version. Finally, they are compared with a proposed local search math‐heuristic algorithm. The proposed models and algorithm are solved with the CPLEX solver with default options. To the best of our knowledge, this journal paper constitutes a first attempt to incorporate network slicing in SDN allowing flexibility, resource efficiency, security, and effective management of the network facilitating the deployment of customized and adaptive services. Besides, our models allow us to deal with the management of connecting users to either controller or switch‐type nodes depending on the slice to which each user belongs. For security reasons, a certain slice could only have access to the network controllers, while the rest of the users that belong to the other slices can connect to the switch‐type nodes of the network. From the numerical experiments, we observe that the linear models show a better performance in terms of CPU times and the best solutions obtained. Similarly, our proposed approximation algorithm achieves near‐optimal solutions in significantly shorter CPU times, for all the input graph networks, when compared to the proposed exact models which allows for finding the optimal solutions.