Linear programming : pivoting on polyhedra and arrangements

Linear programming is perhaps the most useful tool in optimization, much of it's success owed to the efficiency of the simplex method in practice --- its ability to solve problems with millions of variables with relative ease. However, whether there exists a strongly polynomial algorithm to solve linear programming remains an open question. Pivot methods, including the simplex method, remain the best hope for finding such an algorithm, despite the fact that almost all variants have been shown to require exponential time on special instances. Fundamental questions about the path length (number of iterations) of pivot methods remain unanswered. Some, such as the related Hirsch Conjecture, are famous long-standing problems in polyhedral theory. How long can a pivot path be? How many distinct degenerate solutions (bases) can appear during a simplex method cycle? How long can a finite pivot rule stall without improving the solution? Can we enumerate all possible pivot paths to optimality? Can we compute monotone disjoint pivot paths? These are some of the questions we tackle in this thesis in a quest to better understand pivot methods