Upper degree-constrained partial orientations

Abstract

We wish to orient as many edges as possible in an undirected graph (or multigraph), subject to upper bounds on the indegree and out-degree of each vertex. Frank and Gyarfas [2] solve this problem in polynomial time when there are no in-degree bounds, and when every edge can be oriented within the given bounds. However we show that in general the problem is MAXSNP-hard. When viewed as a 3-dimensional matching problem the local improvement algorithm of Hurkens and Schrijver [4] achieves approximation ratio 2/3 -- e; we believe is the best previous bound for our problem. We give an LP-rounding algorithm that achieves approximation ratio 3/4.