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A simple proof that the (n2 − 1)-puzzle is hard

2018; Elsevier BV; Volume: 732; Linguagem: Inglês

10.1016/j.tcs.2018.04.031

1879-2294

Erik D. Demaine, Mikhail Rudoy,

VLSI and FPGA Design Techniques

The 15 puzzle is a classic reconfiguration puzzle with fifteen uniquely labeled unit squares within a $4 \times 4$ board in which the goal is to slide the squares (without ever overlapping) into a target configuration. By generalizing the puzzle to an $n \times n$ board with $n^2-1$ squares, we can study the computational complexity of problems related to the puzzle; in particular, we consider the problem of determining whether a given end configuration can be reached from a given start configuration via at most a given number of moves. This problem was shown NP-complete in Ratner and Warmuth (1990). We provide an alternative simpler proof of this fact by reduction from the rectilinear Steiner tree problem.