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 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, and (ii) the input graph is a tree. Then, by using those algorithms as sub-procedures, we provide a simple, combinatorial, min{max 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.