The Complexity of All-switches Strategy Improvement

TL;DR

This paper proves that determining whether an edge is ever switched or used in the final strategy during all-switches strategy improvement for various types of infinite games is PSPACE-complete, highlighting the computational complexity of these algorithms.

Contribution

It establishes PSPACE-completeness results for key decision problems related to all-switches strategy improvement across multiple game types.

Findings

Edge switch problem is PSPACE-complete for parity, mean-payoff, discounted-payoff, and stochastic games.

Optimal strategy problem is PSPACE-complete for the same game classes.

PSPACE-completeness also holds for the bottom-antipodal algorithm on acyclic orientations.

Abstract

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube.