On Simplex Pivoting Rules and Complexity Theory

Abstract

Abstract. We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path. Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We conjecture that the same can be shown for most known variants of the simplex method. However, we also point out that Dantzig's shadow vertex algorithm has a polynomial path problem. Finally, we discuss in the same context randomized pivoting rules.

The Complexity of the Simplex Method

“…For example, Adler, Papadimitriou, and Rubintstein ask whether such a result holds for all pivoting rules that use only primal feasible bases [2]. Any such recipe would shed light on the requirements for a strongly polynomial-time pivot rule.…”

“…Another recent result along these lines was given by Adler, Papadimitriou and Rubinstein [2], who studied a slightly different problem that they called the path problem: given an LP and a basic feasible solution b, decide whether the simplex method ever visits b. They show that there exists a highly artificial pivot rule for which this problem is in fact PSPACE-complete.…”

Monotone Paths in Geometric Triangulations

“…Let D ⊥ be a set of vectors obtained by rotating each vector in D counterclockwise by π/2 and by −π/2; note that |D ⊥ | = 2|D| ≤ 2|E| ≤ 6n − 12. Sort the vectors in D ⊥ by their arguments 2 in cyclic order, and let U be a set of vector sums of all pairs of consecutive vectors in D ⊥ ; clearly 1 The algorithm has been revised, as some of the ideas were implemented incorrectly in the earlier conference version. 2 The argument of a vector u ∈ R 2 \ {0} is the angle measure in [0, 2π) of the minimum counterclockwise rotation that carries the positive x-axis to the ray spanned by u.…”

Computing Kitahara–Mizuno’s bound on the number of basic feasible solutions generated with the simplex algorithm

“…While over thirty years have passed since the first two polynomial-time algorithms were published [7,9], it is still an open question whether linear programming generally admits a strongly polynomial-time algorithm whose number of iterations depends only on the number of variables n and the number of constraints m. At present, the simplex algorithm is conjectured to serve as a strongly polynomial-time algorithm, by devising an ingenious pivot selection rule [1]. As if to support this, Ye proved in 2011 [16] that the simplex algorithm solves a linear program derived from the Markov decision problem with a fixed discount rate in strongly polynomial time, even under the usual Dantzig's pivot selection rule.…”