Upper and lower degree bounded graph orientation with minimum penalty

Yuichi Asahiro, Jesper Jansson, Eiji Miyano, Hirotaka Ono

研究成果: 書籍/レポート タイプへの寄稿会議への寄与

13 被引用数 (Scopus)


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.

ホスト出版物のタイトルTheory of Computing 2012 - Proceedings of the Eighteenth Computing
ホスト出版物のサブタイトルThe Australasian Theory Symposium, CATS 2012
出版ステータス出版済み - 7月 24 2012
イベントTheory of Computing 2012 - 18th Computing: The Australasian Theory Symposium, CATS 2012 - Melbourne, VIC, オーストラリア
継続期間: 1月 31 20122月 3 2012


名前Conferences in Research and Practice in Information Technology Series


その他Theory of Computing 2012 - 18th Computing: The Australasian Theory Symposium, CATS 2012
CityMelbourne, VIC

!!!All Science Journal Classification (ASJC) codes

  • コンピュータ ネットワークおよび通信
  • コンピュータ サイエンスの応用
  • ハードウェアとアーキテクチャ
  • 情報システム
  • ソフトウェア


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