NxN Puzzle and Related Relocation Problem

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 nxn 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.

[1]  R. J. Schilling,et al.  Decoupling of a Two-Axis Robotic Manipulator Using Nonlinear State Feedback: A Case Study , 1984 .

[2]  Ira Pohl,et al.  Joint and LPA*: Combination of Approximation and Search , 1986, AAAI.

[3]  Paul G. Spirakis,et al.  Coordinating Pebble Motion on Graphs, the Diameter of Permutation Groups, and Applications , 2015, FOCS.

[4]  J. Doran,et al.  Experiments with the Graph Traverser program , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[5]  Judea Pearl,et al.  Heuristics : intelligent search strategies for computer problem solving , 1984 .

[6]  Allen Goldberg,et al.  Is complexity theory of use to AI , 1984 .

[7]  Amos Fiat,et al.  Planning and learning in permutation groups , 1989, 30th Annual Symposium on Foundations of Computer Science.

[8]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[9]  Larry A. Rendell,et al.  A New Basis for State-Space Learning Systems and a Successful Implementation , 1983, Artif. Intell..

[10]  J. Gaschnig Performance measurement and analysis of certain search algorithms. , 1979 .

[11]  Daniel Ratner Issues in theoretical and practical complexity for heuristic search algorithms , 1986 .

[12]  David Lichtenstein,et al.  GO is pspace hard , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[13]  Richard E. Korf,et al.  Depth-First Iterative-Deepening: An Optimal Admissible Tree Search , 1985, Artif. Intell..

[14]  R. Prim Shortest connection networks and some generalizations , 1957 .

[15]  R. Korf Learning to solve problems by searching for macro-operators , 1983 .

[16]  J. Schwartz,et al.  On the Complexity of Motion Planning for Multiple Independent Objects; PSPACE- Hardness of the "Warehouseman's Problem" , 1984 .

[17]  E. Lawler,et al.  Erratum: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1986 .

[18]  Ira Pohl,et al.  The Avoidance of (Relative) Catastrophe, Heuristic Competence, Genuine Dynamic Weighting and Computational Issues in Heuristic Problem Solving , 1973, IJCAI.