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

TL;DR

This paper investigates the existence of Hamiltonian paths in acyclic unique sink orientations of hypercubes under various history-based pivot rules, providing theoretical insights and an enumeration algorithm up to dimension 6.

Contribution

It introduces an enumeration algorithm for Hamiltonian paths in acyclic USOs and analyzes the existence of such paths under different pivot rules across dimensions.

Findings

Zadeh's rule admits Hamiltonian paths up to dimension 9.

Most other rules do not admit Hamiltonian paths beyond dimension 5.

Theoretical and empirical results on the existence of Hamiltonian paths.

Abstract

An acyclic USO on a hypercube is formed by directing its edges in such as 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 modeled 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.