### Abstract

We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective function w. Our preprocessing algorithm replaces w by an integral valued -w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions as w. As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial. Further we apply the preprocessing technique to make H. W. Lenstra's and R. Kannan's Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used. The method relies on simultaneous Diophantine approximation.

Original language | English |
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Pages (from-to) | 49-65 |

Number of pages | 17 |

Journal | Combinatorica |

Volume | 7 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1987 |

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### Keywords

- AMS subject classification (1980): 68E10

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Computational Mathematics

### Cite this

*Combinatorica*,

*7*(1), 49-65. https://doi.org/10.1007/BF02579200

**An application of simultaneous diophantine approximation in combinatorial optimization.** / Frank, A.; Tardos, Éva.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 7, no. 1, pp. 49-65. https://doi.org/10.1007/BF02579200

}

TY - JOUR

T1 - An application of simultaneous diophantine approximation in combinatorial optimization

AU - Frank, A.

AU - Tardos, Éva

PY - 1987/3

Y1 - 1987/3

N2 - We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective function w. Our preprocessing algorithm replaces w by an integral valued -w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions as w. As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial. Further we apply the preprocessing technique to make H. W. Lenstra's and R. Kannan's Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used. The method relies on simultaneous Diophantine approximation.

AB - We present a preprocessing algorithm to make certain polynomial time algorithms strongly polynomial time. The running time of some of the known combinatorial optimization algorithms depends on the size of the objective function w. Our preprocessing algorithm replaces w by an integral valued -w whose size is polynomially bounded in the size of the combinatorial structure and which yields the same set of optimal solutions as w. As applications we show how existing polynomial time algorithms for finding the maximum weight clique in a perfect graph and for the minimum cost submodular flow problem can be made strongly polynomial. Further we apply the preprocessing technique to make H. W. Lenstra's and R. Kannan's Integer Linear Programming algorithms run in polynomial space. This also reduces the number of arithmetic operations used. The method relies on simultaneous Diophantine approximation.

KW - AMS subject classification (1980): 68E10

UR - http://www.scopus.com/inward/record.url?scp=51249175909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249175909&partnerID=8YFLogxK

U2 - 10.1007/BF02579200

DO - 10.1007/BF02579200

M3 - Article

VL - 7

SP - 49

EP - 65

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -