### Abstract

Given an undirected graph with edge weights, we are asked to find an orientation, i.e., an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tight thresholds of complexity: We show that MMO is (i) in P for cactuses, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for P4-bipartite graphs. The latter two are minimal superclasses of the former. Also, we show the NP-hardness for the other related graph classes, diamond-free, house-free, series-parallel, bipartite and planar.

Original language | English |
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Title of host publication | Theory of Computing 2008 - Proceedings of the Fourteenth Computing |

Subtitle of host publication | The Australasian Theory Symposium, CATS 2008 |

Publication status | Published - Dec 1 2008 |

Event | Theory of Computing 2008 - 14th Computing: The Australasian Theory Symposium, CATS 2008 - Wollongong, NSW, Australia Duration: Jan 22 2008 → Jan 25 2008 |

### Publication series

Name | Conferences in Research and Practice in Information Technology Series |
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Volume | 77 |

ISSN (Print) | 1445-1336 |

### Other

Other | Theory of Computing 2008 - 14th Computing: The Australasian Theory Symposium, CATS 2008 |
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Country | Australia |

City | Wollongong, NSW |

Period | 1/22/08 → 1/25/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computer Science Applications
- Hardware and Architecture
- Information Systems
- Software

### Cite this

*Theory of Computing 2008 - Proceedings of the Fourteenth Computing: The Australasian Theory Symposium, CATS 2008*(Conferences in Research and Practice in Information Technology Series; Vol. 77).

**Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree.** / Asahiro, Yuichi; Miyano, Eiji; Ono, Hirotaka.

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

*Theory of Computing 2008 - Proceedings of the Fourteenth Computing: The Australasian Theory Symposium, CATS 2008.*Conferences in Research and Practice in Information Technology Series, vol. 77, Theory of Computing 2008 - 14th Computing: The Australasian Theory Symposium, CATS 2008, Wollongong, NSW, Australia, 1/22/08.

}

TY - GEN

T1 - Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree

AU - Asahiro, Yuichi

AU - Miyano, Eiji

AU - Ono, Hirotaka

PY - 2008/12/1

Y1 - 2008/12/1

N2 - Given an undirected graph with edge weights, we are asked to find an orientation, i.e., an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tight thresholds of complexity: We show that MMO is (i) in P for cactuses, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for P4-bipartite graphs. The latter two are minimal superclasses of the former. Also, we show the NP-hardness for the other related graph classes, diamond-free, house-free, series-parallel, bipartite and planar.

AB - Given an undirected graph with edge weights, we are asked to find an orientation, i.e., an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tight thresholds of complexity: We show that MMO is (i) in P for cactuses, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for P4-bipartite graphs. The latter two are minimal superclasses of the former. Also, we show the NP-hardness for the other related graph classes, diamond-free, house-free, series-parallel, bipartite and planar.

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M3 - Conference contribution

AN - SCOPUS:84863591626

SN - 9781920682583

T3 - Conferences in Research and Practice in Information Technology Series

BT - Theory of Computing 2008 - Proceedings of the Fourteenth Computing

ER -