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Relays in cellular systems are interference limited. The highest end-to-end sum rates are achieved when the relays are jointly optimized with the transmit strategy. Unfortunately, interference couples the links together, making joint optimization challenging. Further, the end-to-end multihop performance is sensitive to rate mismatch when some links have a dominant first link, whereas others have a dominant second link. This paper proposes an algorithm for designing the linear transmit precoders at the transmitters and relays of the relay interference broadcast channel, which is a generic model for relay-based cellular systems, to maximize the end-to-end sum rates. First, the relays are designed to maximize the second-hop sum rates. Next, approximate end-to-end rates that depend on the time-sharing fraction and the second-hop rates are used to formulate a sum-utility maximization problem to design the transmitters. This problem is solved by iteratively minimizing the weighted sum of mean square errors (MSEs). Finally, the norms of the transmit precoders at the transmitters are adjusted to eliminate rate mismatch. The proposed algorithm allows for distributed implementation and has fast convergence. Numerical results show that the proposed algorithm outperforms a reasonable application of single-hop interference management strategies separately on two hops.