Lower bounds for ranking-based pivot rules

TL;DR

This paper establishes superpolynomial and subexponential lower bounds for ranking-based pivot rules in linear programming, strategy improvement, and policy iteration, highlighting fundamental complexity limitations of these methods.

Contribution

It introduces a unified framework for analyzing ranking-based rules and proves significant lower bounds applicable across multiple optimization and decision-making algorithms.

Findings

Superpolynomial lower bound for strategy improvement on sink parity games.

Subexponential lower bound for policy iteration on Markov decision processes.

Results extend to the simplex method for linear programming.

Abstract

The existence of a polynomial pivot rule for the simplex method for linear programming, policy iteration for Markov decision processes, and strategy improvement for parity games each are prominent open problems in their respective fields. While numerous natural candidates for efficient rules have been eliminated, all existing lower bound constructions are tailored to individual or small sets of pivot rules. We introduce a unified framework for formalizing classes of rules according to the information about the input that they rely on. Within this framework, we show lower bounds for \emph{ranking-based} classes of rules that base their decisions on orderings of the improving pivot steps induced by the underlying data. Our first result is a superpolynomial lower bound for strategy improvement, obtained via a family of sink parity games, which applies to memory-based generalizations of Bland's rule that only access the input by comparing the ranks of improving edges in some global order. Our second result is a subexponential lower bound for policy iteration, obtained via a family of Markov decision processes, which applies to memoryless rules that only access the input by comparing improving actions according to their ranks in a global order, their reduced costs, and the associated improvements in objective value. Both results carry over to the simplex method for linear programming.