On the parameterized complexity of the generalized rush hour puzzle

We consider the parameterized complexity of a general- ized version of the game Rush Hour1, which is a puzzle requiring the player to find a sequence of moves by ve- hicles to enable a special target vehicle to escape from a grid-shaped game board that may contain obstacles. Although the problem is PSPACE-complete, we demon- strate algorithms that work in polynomial time when ei- ther the total number of vehicles or the total number of moves is bounded by a constant. Our contributions are two-fold, entailing the application of ideas of param- eterized complexity to games and to motion-planning problems (albeit motion-planning problems of a very constrained nature).