The Worst-Case Complexity of Symmetric Strategy Improvement

TL;DR

This paper demonstrates that symmetric strategy improvement algorithms can have exponential worst-case complexity, providing new worst-case examples that challenge previous assumptions about their efficiency.

Contribution

It introduces a class of worst-case examples for symmetric strategy improvement that exhibit exponential complexity regardless of the improvement rule used.

Findings

Symmetric strategy improvement can take exponential time on certain worst-case instances.

The authors provide a generalized version of the algorithm with similar exponential complexity issues.

Classical strategy improvement instances are often tailored to specific rules, unlike the examples shown here.

Abstract

Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings.