### Abstract

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 V<inf>1</inf> and V<inf>2</inf>, where V<inf>1</inf> contains one of the specified nodes s and V<inf>2</inf> contains the other specified node t, the capacity of a cut is defined as the sum of the edge weights going from V<inf>1</inf> to V<inf>2</inf>. 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 V<inf>1</inf> to V<inf>2</inf> and V<inf>2</inf> to V<inf>1</inf>. 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.

Original language | English (US) |
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Title of host publication | 44th Annual Allerton Conference on Communication, Control, and Computing 2006 |

Publisher | University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering |

Pages | 589-598 |

Number of pages | 10 |

Volume | 2 |

ISBN (Print) | 9781604237924 |

State | Published - 2006 |

Event | 44th Annual Allerton Conference on Communication, Control, and Computing 2006 - Monticello, United States Duration: Sep 27 2006 → Sep 29 2006 |

### Other

Other | 44th Annual Allerton Conference on Communication, Control, and Computing 2006 |
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Country | United States |

City | Monticello |

Period | 9/27/06 → 9/29/06 |

### ASJC Scopus subject areas

- Computer Science Applications
- Computer Networks and Communications

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## Cite this

*44th Annual Allerton Conference on Communication, Control, and Computing 2006*(Vol. 2, pp. 589-598). University of Illinois at Urbana-Champaign, Coordinated Science Laboratory and Department of Computer and Electrical Engineering.