On the existence of Hamiltonian paths for history based pivot rules on acyclic unique sink orientations of hypercubes

Authors:
Yoshikazu Aoshima;David Avis;Theresa Deering;Yoshitake Matsumoto;Sonoko Moriyama
Affiliations:
Department of Computer Science Graduate School of Information Science and Technology, The University of Tokyo, Japan and ERATO-SORST Quantum Computation and Information Project, JST, Japan;School of Informatics, Kyoto University, Japan and School of Computer Science, McGill University, Canada;School of Computer Science, McGill University, Canada;Department of Computer Science Graduate School of Information Science and Technology, The University of Tokyo, Japan and Google Japan Inc., Japan;Graduate School of Information Sciences, Tohoku University, Japan
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

An acyclic USO on a hypercube is formed by directing its edges in such a way that the digraph is acyclic and each face of the hypercube has a unique sink and a unique source. A path to the global sink of an acyclic USO can be modelled as pivoting in a unit hypercube of the same dimension with an abstract objective 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 finding the global sink of an acyclic USO. In this paper we present some theoretical and empirical results on the existence of acyclic USOs for which the various history based pivot rules can be made to follow a Hamiltonian path. In particular, we develop an algorithm that can enumerate all such paths up to dimension 6 using efficient pruning techniques. We show that Zadeh's original rule admits Hamiltonian paths up to dimension 9 at least, and prove that most of the other rules do not for all dimensions greater than 5.