### Abstract

We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A network N=(G, lower, upper) with lower and upper arc capacities is said to be δ-feasible if N has a feasible circulation when we relax the capacity bounds by δ; that is, we use (lower(a)- δ, upper(a)+δ) bounds instead of (lower(a), upper(a)) bounds for each arc a εA. During an approximate binary search we maintain two bounds, LB and UB, such that N is LB-infeasible and UB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph with n vertices, m arcs, and integer capacities bounded by U, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under the similarity condition (that is, U=O(n^{o(1)})). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

Original language | English |
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Pages (from-to) | 243-255 |

Number of pages | 13 |

Journal | Algorithmica |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 1 1994 |

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### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*,

*11*(3), 243-255. https://doi.org/10.1007/BF01240735

**A new scaling algorithm for the maximum mean cut problem.** / Iwano, Kazuo; Misono, Shinji; Tezuka, Shu; Fujishige, Satoru.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 11, no. 3, pp. 243-255. https://doi.org/10.1007/BF01240735

}

TY - JOUR

T1 - A new scaling algorithm for the maximum mean cut problem

AU - Iwano, Kazuo

AU - Misono, Shinji

AU - Tezuka, Shu

AU - Fujishige, Satoru

PY - 1994/3/1

Y1 - 1994/3/1

N2 - We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A network N=(G, lower, upper) with lower and upper arc capacities is said to be δ-feasible if N has a feasible circulation when we relax the capacity bounds by δ; that is, we use (lower(a)- δ, upper(a)+δ) bounds instead of (lower(a), upper(a)) bounds for each arc a εA. During an approximate binary search we maintain two bounds, LB and UB, such that N is LB-infeasible and UB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph with n vertices, m arcs, and integer capacities bounded by U, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under the similarity condition (that is, U=O(no(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

AB - We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A network N=(G, lower, upper) with lower and upper arc capacities is said to be δ-feasible if N has a feasible circulation when we relax the capacity bounds by δ; that is, we use (lower(a)- δ, upper(a)+δ) bounds instead of (lower(a), upper(a)) bounds for each arc a εA. During an approximate binary search we maintain two bounds, LB and UB, such that N is LB-infeasible and UB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph with n vertices, m arcs, and integer capacities bounded by U, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under the similarity condition (that is, U=O(no(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

UR - http://www.scopus.com/inward/record.url?scp=0028401018&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028401018&partnerID=8YFLogxK

U2 - 10.1007/BF01240735

DO - 10.1007/BF01240735

M3 - Article

VL - 11

SP - 243

EP - 255

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 3

ER -