Typical and Extremal Linear Programs

Abstract

Viewed geometrically, the simplex algorithm on a (primally and dually non-degenerate) linear program traces a monotone edge path from the starting vertex to the (unique) optimum. Which path it takes depends on the pivot rule. In this paper we survey geometric and combinatorial aspects of the situation: How do “real” linear programs and their polyhedra look like? How long can simplex paths be in the worst case? Do short paths always exist? Can we expect randomized pivot rules (such as Random Edge) or deterministic rules (such as Zadeh’s rule) to find short paths? MSC 2000. 90C05, 52B11