We study the problem of orienting the edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be N P-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 w max and w min are the maximum and the minimum weights of edges, respectively, and ε is some small positive real number that depends on the input.

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Given an undirected graph with edge weights, we are asked to find an orientation, that is, 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 N P-hard for planar bipartite graphs, and strong N P-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly N P-hard for outerplanar graphs, and also (iii) strongly N P-hard for graphs which are both planar and bipartite. This implies the N P-hardness for P 4-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the N Phardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.