Long Monotone Paths in Abstract Polytopes

Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Mathematics of Operations Research. As is now well known, the simplex method, under its various pivoting rules, is not a "good algorithm" in the sense of J. Edmonds, i.e., the number of pivots needed to solve a given linear programming problem by this method cannot be bounded by a polynomial function of the number of rows and columns defining it. Klee, Minty and Jerosolow have developed methods for constructing examples of such linear programs on polytopes. The aim of this paper is to extend these constructions to abstract polytopes. 1. Introduction. In a recent paper, Klee and Minty [5] constructed a special class of linear programming problems and demonstrated that the simplex method (using certain pivot rules) is not a "good algorithm" in the sense of J. Edmonds. By this is meant that the number of pivots required for solving a given linear program cannot be bounded by a polynomial function of the parameters that determine it, namely the number of rows and columns. They constructed examples requiring a large number of pivots using the usual pivot rules, namely the "random pivot rule" where one moves to any better adjacent vertex, and the "min c rule," where one moves to the adjacent vertex which gives the best per-unit improvement. In a subsequent paper, Jerosolow [4] has shown that the constructive procedure used by Klee and Minty can be used to demonstrate similar behavior under the "best adjacent vertex" rule as well. Abstract polytopes provide a convenient framework for investigating the com-binatorial structure of simple polytopes (nondegenerate bounded systems of linear inequalities). An abstract polytope is a combinatorial system …