K-Center problems with minimum coverage

In this work, we study an extension of the k-center facility location problem, where centers are required to service a minimum of clients. This problem is motivated by requirements to balance the workload of centers while allowing each center to cater to a spread of clients. We study three variants of this problem, all of which are shown to be NP-hard. In-approximation hardness and approximation algorithms with factors equal or close to the best lower bounds are provided. Generalizations, including vertex costs and vertex weights, are also studied.