TY - JOUR
T1 - On the existence of Hamiltonian paths for history based pivot rules on acyclic unique sink orientations of hypercubes
AU - Aoshima, Yoshikazu
AU - Avis, David
AU - Deering, Theresa
AU - Matsumoto, Yoshitake
AU - Moriyama, Sonoko
N1 - Funding Information:
We are grateful to ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency. All of our computational experiments are conducted on the cluster computer in ERATO-SORST. Work on this project was also supported by an INTRIQ-ERATO/SORST collaboration grant funded by MDEIE (Québec) , a discovery grant from NSERC (Canada) , and KAKENHI (Japan) .
Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2012/10
Y1 - 2012/10
N2 - 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.
AB - 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.
KW - Depth first search
KW - Klee-Minty cube
KW - Pivot rule
KW - Polytopal digraphs
KW - Simplex method
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U2 - 10.1016/j.dam.2012.05.023
DO - 10.1016/j.dam.2012.05.023
M3 - Article
AN - SCOPUS:84864776003
VL - 160
SP - 2104
EP - 2115
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
SN - 0166-218X
IS - 15
ER -