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 fifty 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 V₁ and V₂, where V₁ contains one of the specified nodes s and V₂ contains the other specified node t, the capacity of a cut is defined as the sum of the edge weights going from V₁ to V₂. In this paper, we consider a graph partition problem which is encountered in electric power distribution networks. In this environment, the capacity of a cut is defined as the absolute value of the difference of the edge weights going from V₁ to V₂ and V₂ to V₁. 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 definition of the capacity of a cut, the minimum cut computation problem becomes NP-complete. We provide an optimal solution to the problem using mathematical programming techniques. In addition, we also provide a heuristic solution and compare the performance of the optimal solution with the heuristic solution. We also consider a few other closely related problems.