Discussions on Hirsch Conjecture and The Existence of Strongly Polynomial: Time Simplex Variants

Based on the theory of cone-cutting, a preliminary analysis for Hirsch conjecture is made in the paper. The author indirectly proves a related assembling conjecture under condition ‘Fully cuts’ and suggests a relaxed conjecture to deny the existence of super polynomial diameters on LP polytopes. An algorithm named Gradient Falling searching the dual optimal point y* in dual space Y from a given feasible point is presented, which traces a shortest path within the dual feasible region D, the author guesses, it may be an example for the existence of strongly polynomial-times LP algorithms. © 2014, Springer-Verlag GmbH Berlin Heidelberg.