Another Sub-exponential Algorithm for the Simple Stochastic Game

Abstract

We study the problem of solving simple stochastic games, and give both an interesting new algorithm and a hardness result. We show a reduction from fine approximation of simple stochastic games to coarse approximation of a polynomial sized game, which can be viewed as an evidence showing the hardness to approximate the value of simple stochastic games. We also present a randomized algorithm that runs in $\tilde{O}(\sqrt{|V_{\mathrm{R}}|!})$ time, where |V _R| is the number of RANDOM vertices and $\tilde{O}$ ignores polynomial terms. This algorithm is the fastest known algorithm when |V _R|=ω(log n) and $|V_{\mathrm{R}}|=o(\sqrt{\min{|V_{\min}|,|V_{\max}|}})$ and it works for general (non-stopping) simple stochastic games.