Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly ll understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood.  show that in the AUSO setting, Random-Edge is slor than Random-Facet. To do that,  construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2^{Omega(sqrt(n*log(n)))} steps. This improves on a 2^{Omega(sqrt^3(n))} lor bound of Matousek and Szabo. As Random-Facet performs an expected number of 2^{O(sqrt(n)} steps on any n-dimensional AUSO, this established our result. Improving our 2^{Omega(sqrt(n*log(n)))} lor bound seems to require radically new techniques.
['Thomas Dueholm Hansen', 'Uri Zwick']
I am writing a research paper on a idea Random-Edge Is Slower Than Random-Facet on Abstract Cubes.
[{"from": "human", "value": "I am writing a research paper on a idea Random-Edge Is Slower Than Random-Facet on Abstract Cubes."}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly ll understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood.  show that in the AUSO setting, Random-Edge is slor than Random-Facet. To do that,  construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2^{Omega(sqrt(n*log(n)))} steps. This improves on a 2^{Omega(sqrt^3(n))} lor bound of Matousek and Szabo. As Random-Facet performs an expected number of 2^{O(sqrt(n)} steps on any n-dimensional AUSO, this established our result. Improving our 2^{Omega(sqrt(n*log(n)))} lor bound seems to require radically new techniques."}]