The ( n 2-1 )-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 15puzzle to an n xn 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 .beards that is guaranteed to use no more than a constant times the minimum number of moves, where the constant is independent of the given beards 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 robeties 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]  Richard E. Korf,et al.  Depth-First Iterative-Deepening: An Optimal Admissible Tree Search , 1985, Artif. Intell..

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

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

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

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

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

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

[8]  N. Biggs.  THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

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

[10]  E. J.,et al.  ON THE COMPLEXITY OF MOTION PLANNING FOR MULTIPLE INDEPENDENT OBJECTS ; PSPACE HARDNESS OF THE " WAREHOUSEMAN ' S PROBLEM " . * * ) , 2022 .

[11]  T. R. Berger,et al.  Algebra: Groups, rings, and other topics , 1977 .

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

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

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

[15]  David S. Johnson,et al.  The complexity of checkers on an N × N board , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

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

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

[18]  George Politowski.  On the construction of heuristic functions , 1986 .

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