Measures of Intrinsic Hardness for Constraint Satisfaction Problem Instances
Abstract
Our aim is to investigate the factors which determine the intrinsic hardness of constructing a solution to any particular constraint satisfaction problem instance, regardless of the algorithm employed. The line of reasoning is roughly the following: There exists a set of distinct, possibly overlapping, trajectories through the states of the search space, which start at the unique initial state and terminate at complete feasible assignments. These trajectories are named solution paths. The entropy of the distribution of solution paths among the states of each level of the search space provides a measure of the amount of choice available for selecting a solution path at that level. This measure of choice is named solution path diversity. Intrinsic instance hardness is identified with the deficit in solution path diversity and is shown to be linked to the distribution of instance solutions as well as constrainedness, an established hardness measure.
Keywords
Search Space Problem Instance Solution Path Constructive Algorithm Path DiversityPreview
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References
- 1.Williams, C., Hogg, T.: Exploiting the Deep Structure of Constraint Problems. Artificial Intellingence 70, 73–117 (1994)zbMATHCrossRefGoogle Scholar
- 2.Gent, I.P., MacIntyre, E., Prosser, P., Walsh, T.: The Constrainedness of Search. In: AAAI/IAAI, vol. 1, pp. 246–252 (1996)Google Scholar
- 3.Parkes, A.J.: Clustering at the Phase Transition. In: AAAI/IAAI, pp. 340–345 (1997)Google Scholar
- 4.Shannon, C.E.: A Mathematical Theory of Communication. The Bell Systems Technical Journal 27 (1948), Reprinted with correctionsGoogle Scholar
- 5.Slaney, J.: Is there a Constrainedness Knife-Edge? In: Proceedings of the 14th European Conference on Artificial Intelligence, pp. 614–618 (2000)Google Scholar
- 6.Hogg, T., Huberman, B., Williams, C.: Phase Transitions and the Search Problem. Artificial Intelligence 81, 1–15 (1996)CrossRefMathSciNetGoogle Scholar
- 7.Crutchfield, J., Feldman, D.: Regularities Unseen, Randomness Observed: Levels of Entropy Convergence. Chaos 13, 25–54 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
- 8.Vlasie, D.: The Very Particular Structure of the Very Hard Instances. In: AAAI/IAAI, vol. 1, pp. 266–270 (1996)Google Scholar
- 9.Ornstein, D.: Bernoulli Shifts with the Same Entropy are Isomorphic. Adv. in Math. 4, 337–352 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
- 10.Walsh, T.: The Constrainedness Knife-Edge. In: AAAI/IAAI, pp. 406–411 (1998)Google Scholar
- 11.Mezard, M., Ricci-Tersenghi, F., Zecchina, R.: Alternative Solutions to Diluted p-Spin Models and Xorsat Problems. Journal of Statistical Physics (2002)Google Scholar
- 12.Hogg, T.: Which Search Problems are Random? In: Proceedings of AAAI 1998 (1998)Google Scholar
- 13.Hogg, T.: Refining the Phase Transitions in Combinatorial Search. Artificial Intelligence 81, 127–154 (1996)CrossRefMathSciNetGoogle Scholar