A simple proof that the $(n^2-1)$-puzzle is hard

TL;DR

This paper presents a simplified proof demonstrating that determining reachability within a certain number of moves in the generalized $(n^2-1)$-puzzle is NP-complete, using a reduction from the rectilinear Steiner tree problem.

Contribution

It offers a simpler, alternative proof of the NP-completeness of the generalized puzzle problem through reduction from a known NP-hard problem.

Findings

The problem is NP-complete for the generalized puzzle.

A reduction from the rectilinear Steiner tree problem proves the complexity.

The proof simplifies previous complexity demonstrations.

Abstract

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.