On \(\lambda \)-cent-dians and generalized-center for network design: definitions and properties

In this paper, we extend the notions of \(\lambda \)-cent-dian and generalized-center from Facility Location Theory to the more intricate domain of Network Design. Our focus is on the task of designing a sub-network within a given underlying network while adhering to a budget constraint. This sub-network is intended to efficiently serve a collection of origin/destination pairs of demand. The \(\lambda \)-cent-dian problem studies the balance between efficiency and equity. We investigate the properties of the \(\lambda \)-cent-dian and generalized-center solution networks under the lens of equity, efficiency, and Pareto-optimality. We finally prove that the problems solved here are NP-hard.