### Abstract

We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable flow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path as to maximize the total profit of the satisfied terminal pairs subject to the capacity constraints. Classical UFP, in which demands must be lower than edge capacities, is known to have an O(√m) approximation algorithm. We provide the same result with a strongly polynomial combinatorial algorithm. The extended UFP case is when some demands might be higher than edge capacities. For that case we both improve the current best approximation ratio and use strongly polynomial algorithms. We also use a lower bound to show that the extended case is provably harder than the classical case. The last variant is the bounded UFP where demands are at most K of the minimum edge capacity. Using strongly polynomial algorithms here as well, we improve the currently best known algorithms. Specifically, for K = 2 our results are better than the lower bound for classical UFP thereby separating the two problems.

Original language | English (US) |
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Title of host publication | Integer Programming and Combinatorial Optimization - 8th International IPCO Conference, Proceedings |

Publisher | Springer Verlag |

Pages | 15-29 |

Number of pages | 15 |

Volume | 2081 |

ISBN (Print) | 3540422250, 9783540422259 |

State | Published - 2001 |

Event | 8th International Integer Programming and Combinatorial Optimization Conference, IPCO 2001 - Utrecht, Netherlands Duration: Jun 13 2001 → Jun 15 2001 |

### 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 | 2081 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 8th International Integer Programming and Combinatorial Optimization Conference, IPCO 2001 |
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Country | Netherlands |

City | Utrecht |

Period | 6/13/01 → 6/15/01 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Integer Programming and Combinatorial Optimization - 8th International IPCO Conference, Proceedings*(Vol. 2081, pp. 15-29). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2081). Springer Verlag.

**Strongly polynomial algorithms for the unsplittable flow problem.** / Azar, Yossi; Regev, Oded.

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

*Integer Programming and Combinatorial Optimization - 8th International IPCO Conference, Proceedings.*vol. 2081, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2081, Springer Verlag, pp. 15-29, 8th International Integer Programming and Combinatorial Optimization Conference, IPCO 2001, Utrecht, Netherlands, 6/13/01.

}

TY - GEN

T1 - Strongly polynomial algorithms for the unsplittable flow problem

AU - Azar, Yossi

AU - Regev, Oded

PY - 2001

Y1 - 2001

N2 - We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable flow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path as to maximize the total profit of the satisfied terminal pairs subject to the capacity constraints. Classical UFP, in which demands must be lower than edge capacities, is known to have an O(√m) approximation algorithm. We provide the same result with a strongly polynomial combinatorial algorithm. The extended UFP case is when some demands might be higher than edge capacities. For that case we both improve the current best approximation ratio and use strongly polynomial algorithms. We also use a lower bound to show that the extended case is provably harder than the classical case. The last variant is the bounded UFP where demands are at most K of the minimum edge capacity. Using strongly polynomial algorithms here as well, we improve the currently best known algorithms. Specifically, for K = 2 our results are better than the lower bound for classical UFP thereby separating the two problems.

AB - We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable flow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path as to maximize the total profit of the satisfied terminal pairs subject to the capacity constraints. Classical UFP, in which demands must be lower than edge capacities, is known to have an O(√m) approximation algorithm. We provide the same result with a strongly polynomial combinatorial algorithm. The extended UFP case is when some demands might be higher than edge capacities. For that case we both improve the current best approximation ratio and use strongly polynomial algorithms. We also use a lower bound to show that the extended case is provably harder than the classical case. The last variant is the bounded UFP where demands are at most K of the minimum edge capacity. Using strongly polynomial algorithms here as well, we improve the currently best known algorithms. Specifically, for K = 2 our results are better than the lower bound for classical UFP thereby separating the two problems.

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M3 - Conference contribution

SN - 3540422250

SN - 9783540422259

VL - 2081

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

SP - 15

EP - 29

BT - Integer Programming and Combinatorial Optimization - 8th International IPCO Conference, Proceedings

PB - Springer Verlag

ER -