Journal of Artificial Intelligence Research, 7 (1997) 47-66. Submitted 10/96; published 9/97

© 1997 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved.

A New Look at the Easy-Hard-Easy Pattern of Combinatorial Search Difficulty

Dorothy L. Mammen, MAMMEN@CS.UMASS.EDU

Computer Science Department

University of Massachusetts

Amherst, MA 01003, U.S.A.

Tad Hogg HOGG@PARC.XEROX.COM

Xerox Palo Alto Research Center

3333 Coyote Hill Road

Palo Alto, CA 94304, U.S.A.

Abstract

The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easy-hard-easy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions.

Contents

1. Introduction

2. Some Classes of Search Problems

2.1 Random CSPs

2.2 Graph Coloring

3. The Easy-Hard-Easy Pattern

3.1 An Example

3.2 An Explanation

4. Search Difficulty and Solvability

4.1 Search Behavior

4.2 Solvable Problems

4.3 Problems With a Fixed Number of Solutions

5. Minimal Unsolvable Subproblems

6. Conclusions

7. Acknowledgements

References

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