Parity Games with Partial Information Played on Graphs of Bounded Complexity

We address the strategy problem for parity games with partial information and observable colors, played on finite graphs of bounded graph complexity. We consider several measures for the complexity of graphs and analyze in which cases, bounding the measure decreases the complexity of the strategy problem on the corresponding classes of graphs. We prove or disprove that the usual powerset construction for eliminating partial information preserves boundedness of the graph complexity. For the case where the partial information is unbounded we prove that the construction does not preserve boundedness of any measure we consider. We also prove that the strategy problem is Exptimehard on graphs with directed path-width at most 2 and Pspace-complete on acyclic graphs. For games with bounded partial information we obtain that the powerset construction, while neither preserving boundedness of entanglement nor of (undirected) tree-width, does preserve boundedness of directed path-width. Furthermore, if tree-width is bounded then DAGwidth of the resulting graph is bounded. Therefore, parity games with bounded partial information, played on graphs with bounded directed path-width or tree-width can be solved in polynomial time.