An Exponential Lower Bound for Zadeh's pivot rule

TL;DR

This paper proves that Zadeh's pivot rule for the Simplex Algorithm has exponential worst-case running time, resolving a long-standing open question in discrete optimization.

Contribution

It provides the first exponential lower bound for Zadeh's rule, using a novel construction related to strategy improvement and policy iteration algorithms.

Findings

Zadeh's rule is exponential in the worst case.

The construction relates to strategy improvement algorithms for parity games.

Exponential lower bounds also apply to Markov Decision Processes.

Abstract

The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural, deterministic pivot rules are known to yield exponential running times, the random-facet rule was shown to have a subexponential running time. For a long time, Zadeh's rule remained the most prominent candidate for the first deterministic pivot rule with subexponential running time. We present a lower bound construction that shows that Zadeh's rule is in fact exponential in the worst case. Our construction is based on a close relation to the Strategy Improvement Algorithm for Parity Games and the Policy Iteration Algorithm for Markov Decision Processes, and we also obtain exponential lower bounds for Zadeh's rule in these contexts.