The Complexity of the Simplex Method

Fearnley, John and Savani, Rahul ORCID: 0000-0003-1262-7831 (2015) The Complexity of the Simplex Method. In: 47th annual ACM Symposium on Theory of Computing (STOC '15), 2015-6-14 - 2015-6-17.

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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 then show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.

Item Type:	Conference or Workshop Item (Unspecified)
Uncontrolled Keywords:	Linear programming, The simplex method, Dantzig's pivot rule, Markov decision processes, Policy Iteration
Depositing User:	Symplectic Admin
Date Deposited:	01 Feb 2017 10:52
Last Modified:	19 Jan 2023 07:19
DOI:	10.1145/2746539.2746558
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3005481