The (n2−1)-puzzle and related relocation problems

The 8-puzzle and the 15-puzzle have been used for many years as a domain for testing heuristic search techniques. From experience it is known that these puzzles are “difficult” and therefore useful for testing search techniques. In this paper we give strong evidence that these puzzles are indeed good test problems. We extend the 8-puzzle and the 15-puzzle to an n×n board and show that finding a shortest solution for the extended puzzle is NP-hard and is thus believed to be computationally infeasible.We also sketch an approximation algorithm for transforming boards that is guaranteed to use no more than a constant times the minimum number of moves, where the constant is independent of the given boards and their side length n.The studied puzzles are instances of planar relocation problems where the reachability question is polynomial but efficient relocation is NP-hard. Such problems are natural robotics problems: A robot needs to efficiently relocate packages in the plane. Our research encourages the study of polynomial approximation algorithms for related robotics problems