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Current works on the joint routing and channel assignment (JRC) problem in wireless mesh networks (WMNs) assume that all links operate at the base rate. In other words, they do not consider the presence of multiple bit-rates. However, in multi-rate WMNs, the achievable throughput of a link operating at higher rates could be much less than its bit-rate when it has to contend with low-rate links. This is also known as the multirate sharing problem. To address the problem, we first present a numerical formulation for estimating the dynamic link capacity, which can indicate the degree of the multi-rate sharing problem. Next, we propose an optimization model called MR2C, which solves the JRC problem by considering the dynamic link capacity in order to maximize the throughput in multi-rate, multi-radio, multi-channel WMNs. We then reformulate the primal problem as a Lagrangian dual problem, and apply the subgradient method to approach the dual. Finally, since each subproblem in the dual problem is also a non-linear problem, we design a polynomialtime algorithm to solve it. Our experiment results demonstrate that the proposed capacity formulation can approximate the actual link throughput. We compare MR2C with the base-rate JRC model via simulations. The results show that MR2C achieves throughput gains of 3-4 times and recover from network failure efficiently.