Motion planning with pull moves

Sokoban is a game on an integer grid, where a warehouse keeper, or robot, has to push boxes to designated storage locations. He can push a box one cell horizontally or vertically if the destination cell is free. Some of the grid cells can be occupied by unmovable obstacles. Several variants of Sokoban have been studied. In these variants all obstacles may be movable, the keeper may push up to k boxes, or even an unlimited number; to solve the game it can be sufficient to move the keeper to some goal position, and the boxes once pushed may slide until hitting an obstacle. Table 1 shows the complexity of some of these variants. In the table the game type indicates the possible movements. In “Push” games, each box moves one unit, while in “PushPush” a box slides until it hits the next obstacle. The following number indicates the number of boxes which can be moved at once, where “*” stands for infinity. The game type has a suffix “-F” when fixed obstacles are allowed. The classic Sokoban game has game type Push-1-F. Wilfong [13] shows that motion planning where the robot can push and pull polygonal obstacles is NP-hard if we want to decide if the robot can reach a goal position, and PSPACE-complete when goal positions for the obstacles are given. Dor and Zwick [8] show that Sokoban is also PSPACE-complete when the robot can push up to two boxes or pull one box, and the boxes have size 2 × 1. A version of Sokoban where the robot can push as well as pull the boxes is available under the name Pukoban [2]. Polishchuk [12] studies the optimization version of these problems where we want to decide if there exists a solution in less than k box moves (the instances are all trivially solvable). He shows that all of the above variations with designated storage locations and a variation where the robot additionally can lift any number of boxes are NP-hard.