Improved Subexponential Analysis of the Random-Action-Removal Algorithm for 2-Player Turn-Based Games and Non-Binary AUSOs

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 Matoušek, Sharir and Welzl (SoCG 1992). For LP problems, the algorithm of Kalai is a primal simplex algorithm, while the algorithm of Matoušek, Sharir and Welzl is a dual simplex algorithm. The Random-Action-Removal algorithm for games or AUSOs is an adaptation of the dual algorithm of Matoušek, Sharir and Welzl, and is a randomized strategy iteration algorithm. Our improved analysis shows that the Random-Action-Removal algorithm solves games with n states and m≥ 2n actions in e^O(√(nln(m/n))) time. This improves on a previous e^O(√(nln(m/√(n)))) bound for the algorithm that follows from the analysis of Matoušek, Sharir and Welzl (SoCG 1992). An e^O(√(nln(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).