### Abstract

Original language | English |
---|---|

Pages (from-to) | 159-187 |

Journal | Computational Optimization and Applications |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2018 |

### Fingerprint

### Keywords

- vertex separator
- minimum cut
- semidefinite programming
- convexication

### Cite this

*Computational Optimization and Applications*,

*69*(1), 159-187. https://doi.org/10.1007/s10589-017-9943-4

}

*Computational Optimization and Applications*, vol. 69, no. 1, pp. 159-187. https://doi.org/10.1007/s10589-017-9943-4

**The min-cut and vertex separator problem.** / Rendl, Franz; Sotirov, Renata.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - The min-cut and vertex separator problem

AU - Rendl, Franz

AU - Sotirov, Renata

PY - 2018/1

Y1 - 2018/1

N2 - We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n+1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size ( n≈300 ) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying non-convex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.

AB - We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n+1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size ( n≈300 ) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying non-convex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.

KW - vertex separator

KW - minimum cut

KW - semidefinite programming

KW - convexication

U2 - 10.1007/s10589-017-9943-4

DO - 10.1007/s10589-017-9943-4

M3 - Article

VL - 69

SP - 159

EP - 187

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 1

ER -