Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle Is NP-Hard

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Following Wilson (J. Comb. Th. (B), 1975), Johnson (J. of Alg., 1983), and Kornhauser, Miller and Spirakis (25th FOCS, 1984), we consider a game that consists of moving distinct pebbles along the edges of an undirected graph. At most one pebble may reside in each vertex at any time, and it is only allowed to move one pebble at a time (which means that the pebble must be moved to a previously empty vertex). We show that the problem of finding the shortest sequence of moves between two given “pebble configuations” is NP-Hard.