Abstract

We prove NP-hardness of a wide class of pushing-block puzzles similar to the classic Sokoban, generalizing several previous results [E.D. Demaine et al., in: Proc. 12th Canad. Conf. Comput. Geom., 2000, pp. 211–219; E.D. Demaine et al., Technical Report, January 2000; A. Dhagat, J. O'Rourke, in: Proc. 4th Canad. Conf. Comput. Geom., 1992, pp. 188–191; D. Dor, U. Zwick, Computational Geometry 13 (4) (1999) 215–228; J. O'Rourke, Technical Report, November 1999; G. Wilfong, Ann. Math. Artif. Intell. 3 (1991) 131–150]. 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 ( -1) up to arbitrarily many ( -*). Other variations were introduced to make puzzles more tractable, in which blocks must slide their maximal extent when pushed ( ), and in which the robot's path must not revisit itself ( -X). We prove that all of these puzzles are NP-hard.

Author Keywords: Motion planning; Combinatorial games; Computational complexity