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Reconfigurable intelligent surfaces have emerged as a promising technology for future wireless networks. Given that a large number of reflecting elements is typically used, and that the surface has no signal processing capabilities, a major challenge is to cope with the overhead that is required to estimate the channel state information and to report the optimized phase shifts to the surface. This issue has not been addressed by previous works, which do not explicitly consider the overhead during the resource allocation phase. This work aims at filling this gap, developing an overhead-aware resource allocation framework for wireless networks where reconfigurable intelligent surfaces are used to improve the communication performance. An overhead model is developed and incorporated in the expressions of the system rate and energy efficiencies, which are then optimized with respect to the phase shifts of the reconfigurable intelligent surface, the transmit and receive filters, and the power and bandwidth used for the communication and feedback phases. The bi-objective maximization of the rate and energy efficiency is carried out as well. The proposed framework allows characterizing the trade-off between optimized radio resources and the related overhead in networks with reconfigurable intelligent surfaces.