### Abstract

Given an undirected and edge-weighted graph G together with a set of ordered vertex-pairs, called st-pairs, we consider the problems of finding an orientation of all edges in G: min-sum orientation is to minimize the sum of the shortest directed distances between all st-pairs; and min-max orientation is to minimize the maximum shortest directed distance among all st-pairs. In this paper, we first show that both problems are strongly NP-hard for planar graphs even if all edge-weights are identical, and that both problems can be solved in polynomial time for cycles. We then consider the problems restricted to cacti, which form a graph class that contains trees and cycles but is a subclass of planar graphs. Then, min-sum orientation is solvable in polynomial time, whereas min-max orientation remains NP-hard even for two st-pairs. However, based on LP-relaxation, we present a polynomial-time 2-approximation algorithm for min-max orientation. Finally, we give a fully polynomial-time approximation scheme (FPTAS) for min-max orientation on cacti if the number of st-pairs is a fixed constant.

Original language | English |
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Title of host publication | Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings |

Pages | 403-412 |

Number of pages | 10 |

DOIs | |

Publication status | Published - Dec 1 2009 |

Event | 20th International Symposium on Algorithms and Computation, ISAAC 2009 - Honolulu, HI, United States Duration: Dec 16 2009 → Dec 18 2009 |

### 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 | 5878 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 20th International Symposium on Algorithms and Computation, ISAAC 2009 |
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Country | United States |

City | Honolulu, HI |

Period | 12/16/09 → 12/18/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings*(pp. 403-412). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5878 LNCS). https://doi.org/10.1007/978-3-642-10631-6_42

**Route-enabling graph orientation problems.** / Ito, Takehiro; Miyamoto, Yuichiro; Ono, Hirotaka; Tamaki, Hisao; Uehara, Ryuhei.

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

*Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5878 LNCS, pp. 403-412, 20th International Symposium on Algorithms and Computation, ISAAC 2009, Honolulu, HI, United States, 12/16/09. https://doi.org/10.1007/978-3-642-10631-6_42

}

TY - GEN

T1 - Route-enabling graph orientation problems

AU - Ito, Takehiro

AU - Miyamoto, Yuichiro

AU - Ono, Hirotaka

AU - Tamaki, Hisao

AU - Uehara, Ryuhei

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Given an undirected and edge-weighted graph G together with a set of ordered vertex-pairs, called st-pairs, we consider the problems of finding an orientation of all edges in G: min-sum orientation is to minimize the sum of the shortest directed distances between all st-pairs; and min-max orientation is to minimize the maximum shortest directed distance among all st-pairs. In this paper, we first show that both problems are strongly NP-hard for planar graphs even if all edge-weights are identical, and that both problems can be solved in polynomial time for cycles. We then consider the problems restricted to cacti, which form a graph class that contains trees and cycles but is a subclass of planar graphs. Then, min-sum orientation is solvable in polynomial time, whereas min-max orientation remains NP-hard even for two st-pairs. However, based on LP-relaxation, we present a polynomial-time 2-approximation algorithm for min-max orientation. Finally, we give a fully polynomial-time approximation scheme (FPTAS) for min-max orientation on cacti if the number of st-pairs is a fixed constant.

AB - Given an undirected and edge-weighted graph G together with a set of ordered vertex-pairs, called st-pairs, we consider the problems of finding an orientation of all edges in G: min-sum orientation is to minimize the sum of the shortest directed distances between all st-pairs; and min-max orientation is to minimize the maximum shortest directed distance among all st-pairs. In this paper, we first show that both problems are strongly NP-hard for planar graphs even if all edge-weights are identical, and that both problems can be solved in polynomial time for cycles. We then consider the problems restricted to cacti, which form a graph class that contains trees and cycles but is a subclass of planar graphs. Then, min-sum orientation is solvable in polynomial time, whereas min-max orientation remains NP-hard even for two st-pairs. However, based on LP-relaxation, we present a polynomial-time 2-approximation algorithm for min-max orientation. Finally, we give a fully polynomial-time approximation scheme (FPTAS) for min-max orientation on cacti if the number of st-pairs is a fixed constant.

UR - http://www.scopus.com/inward/record.url?scp=75649091880&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=75649091880&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-10631-6_42

DO - 10.1007/978-3-642-10631-6_42

M3 - Conference contribution

AN - SCOPUS:75649091880

SN - 3642106307

SN - 9783642106309

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

SP - 403

EP - 412

BT - Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings

ER -