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Network coding is a promising generalization of routing which allows a node to generate output messages by encoding its received messages. A typical scenario where network coding offers unique advantages is a multicast network where a source node generates messages and multiple receivers collect the messages. In a multicast network, linear network codes are preferred due to its sufficiency and simplicity. In this paper, we propose an approach to transforming the linear coding problem into a graph theory problem. By utilizing hypergraphs, we model the linear codes by constructing a pseudodual graph of the multicast network. Then, a valid linear code is equivalent to a cover in the pseudodual graph satisfying some constraints. By iterative refinements, an eligible cover can be found in polynomial time. Moreover, we propose several preprocessing algorithms to further reduce the computation time required by the iterative refinements by reducing the graph size before transformation. An important contribution of this work is that the proposed approach can be readily extended to solve many minimal network coding problems. By assigning different weights to edges, minimal network coding problems are reduced to the shortest path problem in the pseudodual graph. Our simulation results show that the proposed preprocessing algorithms can reduce the computation time by about 40-50 percent in a medium size multicast network compared to the scheme without preprocessing algorithms, and the throughput of the system with network coding is 25 percent higher than that with the traditional approach of multiple multicast trees.