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 \( d^{{\Theta {\left( {2^{d} } \right)}}} \).

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Correspondence to Jiří Matoušek.

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* This research was partially supported by ETH Zürichan d done in part during the workshop “Towards the Peak” in La Claustra, Switzerland and during a visit to ETH Zürich.

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Matoušek, J. The Number Of Unique-Sink Orientations of the Hypercube*. Combinatorica 26, 91–99 (2006).

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Mathematics Subject Classification (2000):

  • 05C30
  • 90C60