Improved subexponential analysis of the Random-Action-Removal algorithm for 2-player turn-based games and non-binary AUSOs

2026-07-07

AI Summary

This work addresses two-player zero-sum turn-based games defined on finite or infinite-horizon stochastic or deterministic graphs, as well as the problem of finding the unique sink in acyclic unique sink orientations (AUSOs) on non-binary hypercubes. Building upon the Matoušek–Sharir–Welzl Random-Facet algorithmic framework, the study proposes and analyzes a simplified randomized action-removal strategy iteration method, termed Random-Action-Removal. For the first time, this approach achieves an improved subexponential running time bound of $e^{O(\sqrt{n \ln(m/n)})}$, surpassing the previous bound of $e^{O(\sqrt{n \ln(m/\sqrt{n})})}$. The algorithm currently stands as the fastest known randomized method for solving both classes of problems, effectively integrating techniques from strategy iteration, LP-type theory, and combinatorial optimization.

Abstract

We give a concise description and an improved analysis of the Random-Action-Removal algorithm for solving 2-player, 0-sum, turn-based, possibly infinite duration, stochastic or non-stochastic games played on graphs, or on finite sets of states. More generally, the algorithm can be used to find the sink of an Acyclic Unique Sink Orientation (AUSO) of a non-binary hypercube. The families of games that can be solved by the algorithm include discounted and non-discounted stochastic games (SGs) and Mean Payoff Games (MPGs). The obtained algorithm is the fastest known randomized algorithm for solving such games, slightly improving on a much more complicated algorithm of Hansen and Zwick (STOC 2015). The Random-Action-Removal algorithm is an adaptation of the Random-Facet algorithm used to solve linear programming (LP) problems, or, more generally, LP-type problems. Two dual variants of the Random-Facet algorithm were developed independently by Kalai (STOC 1992) and by Matou{š}ek, Sharir and Welzl (SoCG 1992). For LP problems, the algorithm of Kalai is a primal \emph{simplex} algorithm, while the algorithm of Matou{š}ek, Sharir and Welzl is a dual \emph{simplex} algorithm. The Random-Action-Removal algorithm for games or AUSOs is an adaptation of the dual algorithm of Matou{š}ek, Sharir and Welzl, and is a randomized \emph{strategy iteration} algorithm. Our improved analysis shows that the Random-Action-Removal algorithm solves games with~$n$ states and $m\ge 2n$ actions in $e^{O(\sqrt{n\ln(m/n)})}$ time. This improves on a previous $e^{O(\sqrt{n\ln(m/\sqrt n)})}$ bound for the algorithm that follows from the analysis of Matou{š}ek, Sharir and Welzl (SoCG 1992). An $e^{O(\sqrt{n\ln(m/n)})}$ bound, with worse constant factors, was previously obtained using a much more complicated algorithm for solving LP and LP-type problems of Hansen and Zwick (STOC 2015).

Problem

• 2-player turn-based games
• Acyclic Unique Sink Orientation
• Mean Payoff Games
• stochastic games
• non-binary hypercube

Innovation

• Random-Action-Removal
• subexponential algorithm
• AUSO
• strategy iteration
• two-player games