Routing between a base station and a mobile station is one of the key issues that should be addressed in multi-hop cellular systems. It should also take into account maximization of the radio resource utilization of the entire system, in which the radio resource of each cell must be shared by the base station and relay stations. In this paper, we propose a joint bandwidth allocation and routing scheme for an OFDMA-based multi-hop cellular system. This scheme uses a polynomial time heuristic algorithm called Multi-Dimensional Multi-choice Knapsack Problem (MMKP) in order to find an approximate solution maximizing the total downlink throughput. In the simulation results, we show that the proposed scheme finds a sub-optimal solution which is superior to a link quality-based routing scheme, but slightly worse than the optimal solution.