Graph orientation algorithms to minimize the maximum outdegree

This paper studies the problem of orienting all 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 AP-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, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, min{w _max/w_min, (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.