Errata for: A subexponential lower bound for the Random Facet algorithm for Parity Games

TL;DR

This paper corrects a previous claim about the expected number of steps in the Random-Facet algorithm for parity games, providing counterexamples that show the two variants do not have the same expected complexity.

Contribution

It clarifies the relationship between Random-Facet and its permutation-based variant, correcting a prior misconception and providing simple counterexamples.

Findings

Random-Facet and Random-Facet^* have different expected step counts.

Counterexamples demonstrate the false equivalence of the two algorithms.

The previous lower bounds do not directly apply to Random-Facet.

Abstract

In Friedmann, Hansen, and Zwick (2011) we claimed that the expected number of pivoting steps performed by the Random-Facet algorithm of Kalai and of Matousek, Sharir, and Welzl is equal to the expected number of pivoting steps performed by Random-Facet^*, a variant of Random-Facet that bases its random decisions on one random permutation. We then obtained a lower bound on the expected number of pivoting steps performed by Random-Facet^* and claimed that the same lower bound holds also for Random-Facet. Unfortunately, the claim that the expected numbers of steps performed by Random-Facet and Random-Facet^* are the same is false. We provide here simple examples that show that the expected numbers of steps performed by the two algorithms are not the same.