: The examples published by Klee and Minty in 1972 do not preclude the existence of a pivot rule which will make the simplex method, at worst, polynomial. In fact, the continuing success of Dantzig's method suggests that such a rule does exist. A study of known examples shows that (a) those which use 'selective' pivot rules require exponentially large coefficients, and (b) none of the examples' pivot rules are typically used in practice, either because of computational requirements or due to a lack of even-handed movement through the column set. In all 'bad' problems, certain improving columns are entered about 2 to the m-2 power times before other improving columns are entered once. This is done by making the unused columns 'appear' to yield small objective function improvement. The purpose of this paper is to explain the Klee-Minty and Jeroslow constructions, show how they can be modified to be pathological with small integral coefficients, and then suggest a 'least entered' pivot rule which forces an improving column to be entered before any other column is entered for the second time. This rule seems immune to the 'deformed product construction' which is the essence of all known exponential counterexamples. (Author)

What is the Worst Case Behavior of the Simplex Algorithm

This paper reports on an experimental study of the distribution of the length of simplex paths for the Optimal Assignment Problem. We study the distribution of the pivot counts for a version of the simplex method that with essentially equal probabilities introduces any variable with negative reduced cost into the basis.

In this situation the distribution of the pivot counts turns out to be normally distributed and independent of the actual cost coefficients, provided these are sufficiently spread out. Further, the mean and standard deviation grow only moderately with the size of the problem, namely asd1.8, andd1.5 respectively for ad×d problem, implying in particular that the pivot counts concentrate around the mean with growingd.

The usual simplex method on the other hand gives a growth ofd1.6. Hence a large part of the favourable polynomial growth experienced on practical problems may be attributed to the fact that the simplex paths are rather short on the average, at least for assignment problems.

On the length of simplex paths: The assignment case

Vertex and path attributes of the generalized hypercube with simplex cuts

Warren M. Hirsch posed the conjecture which bears his name in a letter of 1957 to George B. Dantzig. Simply stated in geometric terms, Hirsch proposed that a polytope in dimension dwith n facets admits a path of at most (n−d) edges connecting any two vertices. Hirsch posed his conjecture in the context of a linear program in d variables and n constraints as requiring a maximum of (n− d) pivots — steps of the simplex algorithm— on the shortest path, to achieve an optimum. Over the years theHirsch conjecture has attractedmuch research attention, and has been proved in a number of special cases. This article contributes a proof in the general bounded case.

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A constructive inductive proof of the hirsch conjecture

Computational results are presented which show that N. Zadeh's pathological examples for the simplex algorithm apparently take a number of pivots approximately proportional to the number of columns in the tableau when its column order is randomized.

William P. Niedringhaus

Papers

Some experiments with the pathological linear programs of N. Zadeh