Extending Partial Suborders

We consider the problem of finding a transitive orientation T of a comparability graph G = (V, E), such that a given partial order P is extended. Existing algorithms for this problem require the full knowledge of E, so they are of limited use in the context of a branch-and-bound algorithm, where only parts of E may be known at any stage. We present a new approach to the problem by describing a pair of necessary and sufficient conditions for the existence of an orientation T, based on two simple forbidden subconfigurations. This allows it to solve higher-dimensional packing and scheduling problems of interesting size to optimality. We have implemented this approach and the computational results are convincing