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Next-generation wireless networks are preoccupied with the provision of very high data rates in a ubiquitous and fair manner throughout the service area. Toward that end, the deployment of fixed relays by the operators has become an accepted network architecture for which orthogonal frequency-division multiple access (OFDMA) is the envisioned air interface, and efficient resource utilization is imperative. In contrast to the current literature, this paper presents a novel throughput-optimal formulation, which performs joint intracell routing and scheduling, in accordance with the emerging OFDMA-based cellular relay networks employing two-hop half-duplex relaying. Low-complexity iterative algorithms are devised to solve the formulated optimization over two consecutive subframes (the base station transmits, followed by the relay stations) using queue-length coupling. We first show that the network capacity, below which the policy is throughput optimal, has been significantly increased, compared with the previously proposed quasi-full-duplex relaying (FDR) scheme, at a slight complexity increase. Hence, throughput fairness and ubiquity have been improved at high traffic loads, aside from the substantial improvement in both queue-awareness and latency. Second, we show that, without empirical priority weights, our efficient implementation of throughput-optimal scheduling achieves a ubiquitous and fair service within each class of users (with symmetric traffic) and across classes of asymmetric traffic in a relative sense on different time scales. Load balancing among only the active relays could still be jointly realized with the resource allocation.