The Number Of Unique-Sink Orientations of the Hypercube*

Let Q _d denote the graph of the d-dimensional cube. A unique-sink orientation (USO) is an orientation of Q _d such that every face of Q _n has exactly one sink (vertex of out degree 0); it does not have to be acyclic. USO have been studied as an abstract model for many geometric optimization problems, such as linear programming, finding the smallest enclosing ball of a given point set, certain classes of convex programming, and certain linear complementarity problems. It is shown that the number of USO is $d^{{\Theta {\left( {2^{d} } \right)}}}$.