Graph orientation algorithms to minimize the maximum outdegree

Authors:
Yuichi Asahiro;Eiji Miyano;Hirotaka Ono;Kouhei Zenmyo
Affiliations:
Department of Social Information Systems, Kyushu Sangyo University, Fukuoka, Japan;Department of Systems Innovation and Informatics, Kyushu Institute of Technology, Fukuoka, Japan;Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka, Japan;Department of Systems Innovation and Informatics, Kyushu Institute of Technology, Fukuoka, Japan
Venue:
CATS '06 Proceedings of the 12th Computing: The Australasian Theroy Symposium - Volume 51
Year:
2006

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Abstract

We study the problem of orienting the edges of a weighted graph such that the maximum weighted out-degree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be NP-hard generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, or more generally, a graph with identically weighted edges, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, min{wmax/wmin, (2-ε)}-approximation algorithm for the general case, where wmax and wmin are the maximum and the minimum weights of edges, respectively, and ε is some small positive real number that depends on the input.