The optimal partitioning of networks

The work proposes several partitioning criteria, i.e, the flux cut A, the flux cut B, the cost ratio cut, and one generalized minimum cut. The flux cut B is an extension of the flux cut A. The cost ratio cut generates an optimal partitioning for a linear placement problem. The generalized minimum cut uses a cost function related to a polynomial function of the sizes of two partitioned subsets. A high-order polynomial function tends to generate partitions such that the partitioned subsets equal specified sizes. A simplified case is the ratio cut that is shown to derive the clustering structure of the network. The physical meaning of the cuts is described. The partitioning problems are shown to be strongly related to the communication problems. Several network flow models are constructed. We illustrate relations between the maximum flow solutions and the minimum partitionings. We use linear programming to formulate the proposed maximum flow problems. The duality techniques of linear programming are utilized to derive a relation to the optimal partition solution. Thus, we have identified the cases when the optimal partition solutions can be determined in polynomial time.