### Abstract

An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sum k-paths orientation problem, the input is an undirected graph G and ordered pairs (s _{i} ,t _{i} ), where i ∈ {1,2,...,k}. The goal is to find an orientation of G that minimizes the sum over every i ∈ {1,2,...,k} of the distance from s _{i} to t _{i} . In the min-sum k edge-disjoint paths problem the input is the same, however the goal is to find for every i ∈ {1,2,...,k} a path between s _{i} and t _{i} so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k ≥ 2, the question of NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have been open for more than two decades. We study the complexity of these problems when k = 2. We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.

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) |

Publisher | Springer Verlag |

Pages | 1-11 |

Number of pages | 11 |

Volume | 8447 LNCS |

ISBN (Print) | 9783319080000 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

Event | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 - Sophia Antipolis, France Duration: Sep 5 2013 → Sep 6 2013 |

### Publication series

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

Volume | 8447 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 |
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Country | France |

City | Sophia Antipolis |

Period | 9/5/13 → 9/6/13 |

### Fingerprint

### ASJC Scopus subject areas

- 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. 8447 LNCS, pp. 1-11). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8447 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-08001-7_1

**Min-sum 2-paths problems.** / Fenner, Trevor; Lachish, Oded; Popa, Alexandru.

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. 8447 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8447 LNCS, Springer Verlag, pp. 1-11, 11th International Workshop on Approximation and Online Algorithms, WAOA 2013, Sophia Antipolis, France, 9/5/13. https://doi.org/10.1007/978-3-319-08001-7_1

}

TY - GEN

T1 - Min-sum 2-paths problems

AU - Fenner, Trevor

AU - Lachish, Oded

AU - Popa, Alexandru

PY - 2014

Y1 - 2014

N2 - An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sum k-paths orientation problem, the input is an undirected graph G and ordered pairs (s i ,t i ), where i ∈ {1,2,...,k}. The goal is to find an orientation of G that minimizes the sum over every i ∈ {1,2,...,k} of the distance from s i to t i . In the min-sum k edge-disjoint paths problem the input is the same, however the goal is to find for every i ∈ {1,2,...,k} a path between s i and t i so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k ≥ 2, the question of NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have been open for more than two decades. We study the complexity of these problems when k = 2. We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.

AB - An orientation of an undirected graph G is a directed graph obtained by replacing each edge {u,v} of G by exactly one of the arcs (u,v) or (v,u). In the min-sum k-paths orientation problem, the input is an undirected graph G and ordered pairs (s i ,t i ), where i ∈ {1,2,...,k}. The goal is to find an orientation of G that minimizes the sum over every i ∈ {1,2,...,k} of the distance from s i to t i . In the min-sum k edge-disjoint paths problem the input is the same, however the goal is to find for every i ∈ {1,2,...,k} a path between s i and t i so that these paths are edge-disjoint and the sum of their lengths is minimum. Note that, for every fixed k ≥ 2, the question of NP-hardness for the min-sum k-paths orientation problem and the min-sum k edge-disjoint paths problem have been open for more than two decades. We study the complexity of these problems when k = 2. We exhibit a PTAS for the min-sum 2-paths orientation problem. A by-product of this PTAS is a reduction from the min-sum 2-paths orientation problem to the min-sum 2 edge-disjoint paths problem. The implications of this reduction are: (i) an NP-hardness proof for the min-sum 2-paths orientation problem yields an NP-hardness proof for the min-sum 2 edge-disjoint paths problem, and (ii) any approximation algorithm for the min-sum 2 edge-disjoint paths problem can be used to construct an approximation algorithm for the min-sum 2-paths orientation problem with the same approximation guarantee and only an additive polynomial increase in the running time.

UR - http://www.scopus.com/inward/record.url?scp=84903635435&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903635435&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-08001-7_1

DO - 10.1007/978-3-319-08001-7_1

M3 - Conference contribution

AN - SCOPUS:84903635435

SN - 9783319080000

VL - 8447 LNCS

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

SP - 1

EP - 11

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

PB - Springer Verlag

ER -