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We study the problem of channel capacity, and flow assignment (CCFA) in multi-channel wireless mesh networks (WMNs). CCFA involves the joint assignment of channels, distribution of wireless capacity, and determination of link flows to enhance the effectiveness of WMNs. We first study the capacity assignment (CA) problem in WMNs (WMN-CA) which involves the distribution of wireless capacity, given the topology and the flows (i.e., traffic demands and routing). Unlike wired networks, the capacities of different wireless links in a WMN have to be carved out of the capacities of wireless nodes. Since the wireless medium is shared by various wireless nodes, interference between different wireless links constrains the distribution of the wireless capacity available at individual nodes. We formulate WMN-CA as a convex non-linear optimization problem (NLP). We also present efficient heuristics to solve the problem and compare their relative performance. We then propose a linear programming (LP) based iterative algorithm for CCFA. We define a new metric, called network utility, which takes both throughput and average packet delay into account and includes a parameter that can be used to emphasize one over the other. Our approach has two separate phases - (1) channel assignment and (2) multi-channel capacity and flow assignment. The two phases of the iterative CCFA algorithm are performed repeatedly to improve the network utility which allows for a tradeoff between delay and throughput that can be achieved in a WMN.