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 $\mathcal{NP}$-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, $\min \{\frac{w_{max}}{w_{min}},(2 - \varepsilon)\}$-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. [ABSTRACT FROM AUTHOR]