### Abstract

Given an undirected graph G = (V,E), a graph orientation problem is to decide a direction for each edge so that the resulting directed graph G = (V, Λ(E)) satisfies a certain condition, where Λ(E) is a set of assignments of a direction to each edge {u, v} ∈ E. Among many conceivable types of conditions, we consider a degree constrained orientation: Given positive integers a _{v} and b _{v} for each v (a _{v} ≤ b _{v}), decide an orientation of G so that a _{v} ≤ {pipe}{(v, u) ∈ Λ(E)}{pipe} ≤ b _{v} holds for every v ∈ V. However, such an orientation does not always exist. In this case, it is desirable to find an orientation that best fits the condition instead. In this paper, we consider the problem of finding an orientation that minimizes Σ _{v∈V} ^{cv}, where c _{v} is a penalty incurred for v's violating the degree constraint. As penalty functions, several classes of functions can be considered, e.g., linear functions, convex functions and concave functions. We show that the degree-constrained orientation with any convex (including linear) penalty function can be solved in O(m ^{1:5} min{Δ ^{0:5}, log(nC)}), where n = {pipe}V{pipe},m = {pipe}E{pipe}, Δ and C are the maximum degree and the largest magnitude of a penalty, respectively. In contrast, it has no polynomial approximation algorithm whose approximation factor is better than 1.3606, for concave penalty functions, unless P=NP; it is APX-hard. This holds even for step functions, which are considered concave. For trees, the problem with any penalty functions can be solved exactly in O(n log Δ) time, and if the penalty function is convex, it is solvable in linear time.

Original language | English |
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Title of host publication | Theory of Computing 2012 - Proceedings of the Eighteenth Computing |

Subtitle of host publication | The Australasian Theory Symposium, CATS 2012 |

Pages | 139-146 |

Number of pages | 8 |

Publication status | Published - Jul 24 2012 |

Event | Theory of Computing 2012 - 18th Computing: The Australasian Theory Symposium, CATS 2012 - Melbourne, VIC, Australia Duration: Jan 31 2012 → Feb 3 2012 |

### Publication series

Name | Conferences in Research and Practice in Information Technology Series |
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Volume | 128 |

ISSN (Print) | 1445-1336 |

### Other

Other | Theory of Computing 2012 - 18th Computing: The Australasian Theory Symposium, CATS 2012 |
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Country | Australia |

City | Melbourne, VIC |

Period | 1/31/12 → 2/3/12 |

### 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 2012 - Proceedings of the Eighteenth Computing: The Australasian Theory Symposium, CATS 2012*(pp. 139-146). (Conferences in Research and Practice in Information Technology Series; Vol. 128).

**Upper and lower degree bounded graph orientation with minimum penalty.** / Asahiro, Yuichi; Jansson, Jesper; Miyano, Eiji; Ono, Hirotaka.

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

*Theory of Computing 2012 - Proceedings of the Eighteenth Computing: The Australasian Theory Symposium, CATS 2012.*Conferences in Research and Practice in Information Technology Series, vol. 128, pp. 139-146, Theory of Computing 2012 - 18th Computing: The Australasian Theory Symposium, CATS 2012, Melbourne, VIC, Australia, 1/31/12.

}

TY - GEN

T1 - Upper and lower degree bounded graph orientation with minimum penalty

AU - Asahiro, Yuichi

AU - Jansson, Jesper

AU - Miyano, Eiji

AU - Ono, Hirotaka

PY - 2012/7/24

Y1 - 2012/7/24

N2 - Given an undirected graph G = (V,E), a graph orientation problem is to decide a direction for each edge so that the resulting directed graph G = (V, Λ(E)) satisfies a certain condition, where Λ(E) is a set of assignments of a direction to each edge {u, v} ∈ E. Among many conceivable types of conditions, we consider a degree constrained orientation: Given positive integers a v and b v for each v (a v ≤ b v), decide an orientation of G so that a v ≤ {pipe}{(v, u) ∈ Λ(E)}{pipe} ≤ b v holds for every v ∈ V. However, such an orientation does not always exist. In this case, it is desirable to find an orientation that best fits the condition instead. In this paper, we consider the problem of finding an orientation that minimizes Σ v∈V cv, where c v is a penalty incurred for v's violating the degree constraint. As penalty functions, several classes of functions can be considered, e.g., linear functions, convex functions and concave functions. We show that the degree-constrained orientation with any convex (including linear) penalty function can be solved in O(m 1:5 min{Δ 0:5, log(nC)}), where n = {pipe}V{pipe},m = {pipe}E{pipe}, Δ and C are the maximum degree and the largest magnitude of a penalty, respectively. In contrast, it has no polynomial approximation algorithm whose approximation factor is better than 1.3606, for concave penalty functions, unless P=NP; it is APX-hard. This holds even for step functions, which are considered concave. For trees, the problem with any penalty functions can be solved exactly in O(n log Δ) time, and if the penalty function is convex, it is solvable in linear time.

AB - Given an undirected graph G = (V,E), a graph orientation problem is to decide a direction for each edge so that the resulting directed graph G = (V, Λ(E)) satisfies a certain condition, where Λ(E) is a set of assignments of a direction to each edge {u, v} ∈ E. Among many conceivable types of conditions, we consider a degree constrained orientation: Given positive integers a v and b v for each v (a v ≤ b v), decide an orientation of G so that a v ≤ {pipe}{(v, u) ∈ Λ(E)}{pipe} ≤ b v holds for every v ∈ V. However, such an orientation does not always exist. In this case, it is desirable to find an orientation that best fits the condition instead. In this paper, we consider the problem of finding an orientation that minimizes Σ v∈V cv, where c v is a penalty incurred for v's violating the degree constraint. As penalty functions, several classes of functions can be considered, e.g., linear functions, convex functions and concave functions. We show that the degree-constrained orientation with any convex (including linear) penalty function can be solved in O(m 1:5 min{Δ 0:5, log(nC)}), where n = {pipe}V{pipe},m = {pipe}E{pipe}, Δ and C are the maximum degree and the largest magnitude of a penalty, respectively. In contrast, it has no polynomial approximation algorithm whose approximation factor is better than 1.3606, for concave penalty functions, unless P=NP; it is APX-hard. This holds even for step functions, which are considered concave. For trees, the problem with any penalty functions can be solved exactly in O(n log Δ) time, and if the penalty function is convex, it is solvable in linear time.

UR - http://www.scopus.com/inward/record.url?scp=84864023399&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864023399&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84864023399

SN - 9781921770098

T3 - Conferences in Research and Practice in Information Technology Series

SP - 139

EP - 146

BT - Theory of Computing 2012 - Proceedings of the Eighteenth Computing

ER -