Abstract
In this paper we investigate the computational complexity of combinatorial problems with givens, i.e., partial solutions, and where a unique solution is required. Examples for this article are taken from the games of Sudoku, N-queens and related games. We will show the computational complexity of many decision and search problems related to Sudoku, a number of similar games and their generalization. Furthermore, we propose a logical description of several such problems that can lead to a formulation in the language of Quantified Boolean Formulae (QBF) and, hence, their mechanization via a QBF solver. Some experiments on finding the minimum number of givens necessary/sufficient to guarantee uniqueness of solution are shown.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beame, P., Cook, S., Edmonds, J., Impagliazzo, R., Pitassi, T.: The relative complexity of NP search problems. Journal of Computer and System Sciences 57(1), 3–19 (1998)
Blass, A., Gurevich, Y.: On the unique satisfiability problem. Information and Computation 55(1-3), 80–88 (1982)
Cadoli, M., Schaerf, A.: Compiling problem specifications into SAT. Artificial Intelligence 162, 89–120 (2005)
Cadoli, M., Schaerf, M., Giovanardi, A., Giovanardi, M.: An algorithm to evaluate quantified boolean formulae and its experimental evaluation. Journal of Automated Reasoning 28, 101–142 (2002)
Dechter, R.: Constraint Networks (Survey), pp. 276–285. John Wiley & Sons, Inc., Chichester (1992)
Fagin, R.: Generalized First-Order Spectra and Polynomial-Time Recognizable Sets. In: Karp, R.M. (ed.) Complexity of Computation, pp. 43–74. AMS (1974)
Gent, I.P., Walsh, T.: Csplib: A benchmark library for constraints. Technical report, Technical report APES-09-1999. A shorter version appears in the Proceedings of the 5th International Conference on Principles and Practices of Constraint Programming, CP 1999 (1999), Available from: http://csplib.cs.strath.ac.uk/
Johnson, D.S.: A catalog of complexity classes. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 2, vol. A, pp. 67–161. Elsevier Science Publishers (North-Holland), Amsterdam (1990)
Megiddo, N., Papadimitriou, C.H.: On total functions, existence theorems and computational complexity. Theoretical Computer Science 81, 317–324 (1991)
Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the Thirtyeighth Conference on Design Automation (DAC 2001), pp. 530–535. ACM Press, New York (2001)
Stockmeyer, L.J.: The polynomial-time hierarchy. Theoretical Computer Science 3, 1–22 (1976)
Van Hentenryck, P.: The OPL Optimization Programming Language. The MIT Press, Cambridge (1999)
Yato, T., Seta, T.: Complexity and completeness of finding another solution and its application to puzzles (2002), Available at: http://www.phil.uu.nl/~oostrom/cki20/02-03/japansepuzzles/ASP.pdf
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cadoli, M., Schaerf, M. (2006). Partial Solutions with Unique Completion. In: Stock, O., Schaerf, M. (eds) Reasoning, Action and Interaction in AI Theories and Systems. Lecture Notes in Computer Science(), vol 4155. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11829263_6
Download citation
DOI: https://doi.org/10.1007/11829263_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37901-0
Online ISBN: 978-3-540-37902-7
eBook Packages: Computer ScienceComputer Science (R0)