Pushing blocks is hard

We prove NP-hardness of a wide class of pushing-block puzzles similar to the classic Sokoban, generalizing several previous results [5, 6, 9, 10, 15, 17]. The puzzles consist of unit square blocks on an integer lattice; all blocks are movable. The robot may move horizontally and vertically in order to reach a specified goal position. The puzzle variants differ in the number of blocks that the robot can push at once, ranging from at most one (PUSH-1) up to arbitrarily many (PUSH-*). Other variations were introduced to make puzzles more tractable, in which blocks must slide their maximal extent when pushed (PUSHPUSH), and in which the robot’s path must not revisit itself (PUSH-X). We prove that all of these puzzles are NP-hard.