A new min-cut problem with application to electric power network partitioning

The problem of partitioning a graph into two or more subgraphs that satisfies certain conditions is encountered in many different domains. Accordingly, graph partitioning problem has been studied extensively in the last 50 years. The most celebrated result among this class of problems is the max flow = min cut theorem due to Ford and Fulkerson. Utilizing the modifications suggested by Edmonds and Karp. it is well known that the minimum capacity cut in the directed graph with edge weights can be computed in polynomial time. If the partition divides the node set V into subsets V-1 and V-2, where V-1 contains one of the specified nodes s and V-2 contains the other specified node t, the capacity of a cut is defined as the sum of the edge weights going from V-1 to V-2. In electrical power distribution networks, a slow-coherency-based islanding strategy is used as a prevention against the cascading failures. In this paper, we concentrate on the graph partition problems which are encountered in electric power distribution networks. In this environment, two different definitions of capacity of a cut are used. In the first definition, capacity of a cut is taken to be the difference of the edge weights going from V-1 to V-2 and from V-2 to V-1. In the second definition, the capacity of a cut is taken to be the maximum of sum of the edge weights going front V-1 to V-2 and from V-2 to V-1. Surprisingly, with slight change of the definition of the capacity of a cut, the computational complexity of the problem changes significantly. In this paper, we show that with the new definitions of the capacity of a cut, the minimum cut computation problem becomes NP-complete. We provide an optimal solution to the problems using mathematical programming techniques. In addition, we also provide heuristic solutions and compare the performance with that of the optimal solution. Copyright (C) 2008 John Wiley & Sons, Ltd.