### 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-sumk-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 all i∈{1,2,…,k} of the distance from s_{i} to t_{i}. In the min-sumkedge-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 for the min-sum k edge-disjoint paths problem has 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 |
---|---|

Pages (from-to) | 94-110 |

Number of pages | 17 |

Journal | Theory of Computing Systems |

Volume | 58 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2016 |

Externally published | Yes |

### Fingerprint

### Keywords

- Algorithms
- Graph
- Min-sum
- Orientation
- Paths

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing Systems*,

*58*(1), 94-110. https://doi.org/10.1007/s00224-014-9569-1

**Min-Sum 2-Paths Problems.** / Fenner, Trevor; Lachish, Oded; Popa, Alexandru.

Research output: Contribution to journal › Article

*Theory of Computing Systems*, vol. 58, no. 1, pp. 94-110. https://doi.org/10.1007/s00224-014-9569-1

}

TY - JOUR

T1 - Min-Sum 2-Paths Problems

AU - Fenner, Trevor

AU - Lachish, Oded

AU - Popa, Alexandru

PY - 2016/1/1

Y1 - 2016/1/1

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-sumk-paths orientation problem, the input is an undirected graph G and ordered pairs (si,ti), where i∈{1,2,…,k}. The goal is to find an orientation of G that minimizes the sum over all i∈{1,2,…,k} of the distance from si to ti. In the min-sumkedge-disjoint paths problem, the input is the same, however the goal is to find for every i∈{1,2,…,k} a path between si and ti 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 for the min-sum k edge-disjoint paths problem has 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-sumk-paths orientation problem, the input is an undirected graph G and ordered pairs (si,ti), where i∈{1,2,…,k}. The goal is to find an orientation of G that minimizes the sum over all i∈{1,2,…,k} of the distance from si to ti. In the min-sumkedge-disjoint paths problem, the input is the same, however the goal is to find for every i∈{1,2,…,k} a path between si and ti 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 for the min-sum k edge-disjoint paths problem has 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.

KW - Algorithms

KW - Graph

KW - Min-sum

KW - Orientation

KW - Paths

UR - http://www.scopus.com/inward/record.url?scp=84952944101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84952944101&partnerID=8YFLogxK

U2 - 10.1007/s00224-014-9569-1

DO - 10.1007/s00224-014-9569-1

M3 - Article

VL - 58

SP - 94

EP - 110

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 1

ER -