The Complexity of the Simplex Method

TL;DR

This paper proves that determining the outcome of the simplex method with Dantzig's pivot rule is PSPACE-complete, revealing deep computational complexity aspects of a fundamental linear programming algorithm.

Contribution

It establishes PSPACE-completeness results for the simplex method's solution and pivot decisions, connecting linear programming complexity with Markov decision processes.

Findings

Finding the simplex solution with Dantzig's rule is PSPACE-complete.

Deciding if Dantzig's rule chooses a specific variable is PSPACE-complete.

Uses MDPs and policy iteration to prove complexity results.

Abstract

The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.