We consider a virtualized RAN architecture for 5G networks where the Remote Units are connected to a central unit via a mid-haul. To support high data rates, the mid-haul is realized with a Passive Optical Network (PON). In this architecture, the data are stored at the central unit until the scheduler decides to transmit it through the mid-haul to an appropriate remote unit, and then over the air at the same slot. We study an optimal scheduling problem that arises in this context. This problem has two key features. First, multiple cells must be scheduled simultaneously for efficient operation. Second, the interplay between the time-varying wireless interface rates and the fixed capacity PON needs to be handled efficiently. In this paper, we take a comprehensive look at this resource allocation problem by formulating it as a utility-maximization problem. Using combinatorial techniques, we derive useful structural properties of the optimal allocation and utilize these results to design polynomial-time approximation algorithms and a pseudo- polynomial-time optimal algorithm. Finally, we numerically compare the performance of the proposed algorithms to heuristics which are natural generalizations of the ubiquitous Proportional Fair algorithm. © 2019 IFIP.