  The optimal value computation for turned-based stochastic games with
reachability objectives, also known as simple stochastic games, is one of the
few problems in $NP \cap coNP$ which are not known to be in $P$. However, there
are some cases where these games can be easily solved, as for instance when the
underlying graph is acyclic. In this work, we try to extend this tractability
to several classes of games that can be thought as "almost" acyclic. We give
some fixed-parameter tractable or polynomial algorithms in terms of different
parameters such as the number of cycles or the size of the minimal feedback
vertex set.
