SIAM Journal on Discrete Mathematics
Computing the inertia from sign patterns
IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem
Journal of Combinatorial Theory Series B
Approximation scheme for lowest outdegree orientation and graph density measures
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
Upper and lower degree bounded graph orientation with minimum penalty
CATS '12 Proceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 128
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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 Gyárfás [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 -- ε; we believe is the best previous bound for our problem. We give an LP-rounding algorithm that achieves approximation ratio 3/4.