### Abstract

Several studies have reported that the linear program relaxation of integer multi-commodity network flow problems often provides integer optimal solutions. We explore this phenomenon with a 0-1 multi-commodity network with mutual arc capacity constraints. Characteristics of basic solutions in the linear programming relaxation problem of the 0-1 multi-commodity problem are identified. Specifically, necessary conditions for a linear programming relaxation to have a non-integer solution are presented. Based on the observed characteristics, a simple illustrative example problem is constructed to show that its LP relaxation problem has integer optimal solutions with a relatively high probability. Furthermore, to investigate whether or not and under what conditions this tendency applies to large-sized problems, we have carried out computational experiments by using randomly generated problem instances. The results of our computational experiment indicate that there exists a narrow band of arc density in which the 0-1 multi-commodity problems possess no integer optimal solutions.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 517-526 |

Number of pages | 10 |

Volume | 4112 LNCS |

Publication status | Published - 2006 Oct 16 |

Event | 12th Annual International Conference on Computing and Combinatorics, COCOON 2006 - Taipei, Taiwan, Province of China Duration: 2006 Aug 15 → 2006 Aug 18 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4112 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 12th Annual International Conference on Computing and Combinatorics, COCOON 2006 |
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Country | Taiwan, Province of China |

City | Taipei |

Period | 06/8/15 → 06/8/18 |

### Fingerprint

### ASJC Scopus subject areas

- Biochemistry, Genetics and Molecular Biology(all)
- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 4112 LNCS, pp. 517-526). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4112 LNCS).

**On the effectiveness of the linear programming relaxation of the 0-1 multi-commodity minimum cost network flow problem.** / Choi, Dae S.; Choi, In Chan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 4112 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4112 LNCS, pp. 517-526, 12th Annual International Conference on Computing and Combinatorics, COCOON 2006, Taipei, Taiwan, Province of China, 06/8/15.

}

TY - GEN

T1 - On the effectiveness of the linear programming relaxation of the 0-1 multi-commodity minimum cost network flow problem

AU - Choi, Dae S.

AU - Choi, In Chan

PY - 2006/10/16

Y1 - 2006/10/16

N2 - Several studies have reported that the linear program relaxation of integer multi-commodity network flow problems often provides integer optimal solutions. We explore this phenomenon with a 0-1 multi-commodity network with mutual arc capacity constraints. Characteristics of basic solutions in the linear programming relaxation problem of the 0-1 multi-commodity problem are identified. Specifically, necessary conditions for a linear programming relaxation to have a non-integer solution are presented. Based on the observed characteristics, a simple illustrative example problem is constructed to show that its LP relaxation problem has integer optimal solutions with a relatively high probability. Furthermore, to investigate whether or not and under what conditions this tendency applies to large-sized problems, we have carried out computational experiments by using randomly generated problem instances. The results of our computational experiment indicate that there exists a narrow band of arc density in which the 0-1 multi-commodity problems possess no integer optimal solutions.

AB - Several studies have reported that the linear program relaxation of integer multi-commodity network flow problems often provides integer optimal solutions. We explore this phenomenon with a 0-1 multi-commodity network with mutual arc capacity constraints. Characteristics of basic solutions in the linear programming relaxation problem of the 0-1 multi-commodity problem are identified. Specifically, necessary conditions for a linear programming relaxation to have a non-integer solution are presented. Based on the observed characteristics, a simple illustrative example problem is constructed to show that its LP relaxation problem has integer optimal solutions with a relatively high probability. Furthermore, to investigate whether or not and under what conditions this tendency applies to large-sized problems, we have carried out computational experiments by using randomly generated problem instances. The results of our computational experiment indicate that there exists a narrow band of arc density in which the 0-1 multi-commodity problems possess no integer optimal solutions.

UR - http://www.scopus.com/inward/record.url?scp=33749561172&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749561172&partnerID=8YFLogxK

M3 - Conference contribution

SN - 3540369252

SN - 9783540369257

VL - 4112 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 517

EP - 526

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -