Considering a multihop cellular system with one relay per sector, an effective modeling for the joint base-station/relay assignment, rate allocation, and routing scheme is proposed and formulated under a single problem for the downlink. This problem is then formulated as a multidimensional multichoice knapsack problem (MMKP) to maximize the total achieved throughput in the network. The well-known MMKP algorithm based on Lagrange multipliers is modified, which results in a near-optimal solution with a linear complexity. The notion of the infeasibility factor is also introduced to adjust the transmit power of base stations and relays adaptively. To reduce the complexity, and in order to analyze the underlying key factors in the system, the framework is restricted to a two-base-station two-relay system. In fact, the output of the proposed algorithm is the joint optimization of the routing path, and base-station selection to achieve the maximum total throughput in the system, which in conjunction with the proposed adaptive scheme leads to the implementation of the cell breathing via allocating the proper transmit power to the base-stations and relays.
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