The box mover problem

We show that the optimization problem is NP-hard for a wide class of motion planning puzzles, including classical SOKOBAN. We investigate a new problem, the Box Mover Problem (BMP), in which the agent is allowed to lift and carry boxes on a rectilinear grid in order to rearrange them. Some classical motion planning puzzles are special cases of BMP. We also identify a natural class of BMP instances, for which optimization is in NP, making the optimization problems from the class NP-complete.