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Existing works designed for the multi-radio multi-channel (MR-MC) wireless network mainly rely on the orthogonal or non-overlapping channels. But in reality, the limited number of non-overlapping channels is a major issue when the network is dense. In this paper, we study the impact of partially overlapping channels (POC) on the network capacity through computing the maximum achievable capacity in the MR-MC context. We first extend our tool of multi-dimensional conflict graph (MDCG)  to the weighted MDCG (WMDCG) under the physical interference model, where the weight of each edge accurately indicates the amount of interference. Such a tool facilitates the theoretical analysis considering the POC. We then develop a computing methodology to calculate the optimal achievable network capacity with POC by formulating a linear programming (LP) multi-commodity flow (MCF) problem, augmented by the interference constraints based on the WMDCG. The numerical results demonstrate the effectiveness of our computing methodology.