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An acyclic USO on a hypercube is formed by directing its edges in such as waythat the digraph is acyclic and each face of the hypercube has a unique sinkand a unique source. A path to the global sink of an acyclic USO can be modeledas pivoting in a unit hypercube of the same dimension with an abstractobjective function, and vice versa. In such a way, Zadeh's 'least entered rule'and other history based pivot rules can be applied to the problem of findingthe global sink of an acyclic USO. In this paper we present some theoreticaland empirical results on the existence of acyclic USOs for which the varioushistory based pivot rules can be made to follow a Hamiltonian path. Inparticular, we develop an algorithm that can enumerate all such paths up todimension 6 using efficient pruning techniques. We show that Zadeh's originalrule admits Hamiltonian paths up to dimension 9 at least, and prove that mostof the other rules do not for all dimensions greater than 5.

On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes