Abstract
Conjunctive-query containment is considered as a fundamental problem in database query evaluation and optimization. Kolaitis and Vardi pointed out that constraint satisfaction and conjunctive query containment are essentially the same problem. We study the Boolean conjunctive queries under a more detailed scope, where we investigate their counting problem by means of the algebraic approach through Galois theory, taking advantage of Post’s lattice. We prove a trichotomy theorem for the generalized conjunctive query counting problem, showing this way that, contrary to the corresponding decision problems, constraint satisfaction and conjunctive-query containment differ for other computational goals. We also study the audit problem for conjunctive queries asking whether there exists a frozen variable in a given query. This problem is important in databases supporting statistical queries. We derive a dichotomy theorem for this audit problem that sheds more light on audit applicability within database systems.
Supported by ÉGIDE 05835SH, DAAD D/0205776 and DFG VO 630/5-1.
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References
Abiteboul, S., Hull, R., Vianu, V.: Foundation of databases. Addison-Wesley, Reading (1995)
Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory. SIGACT News, Complexity Theory Column 42, 34(4), 38–52 (2003)
Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, part II: Constraint satisfaction problems. SIGACT News, Complexity Theory Column 43, 35(1), 22–35 (2004)
Creignou, N., Hermann, M.: On #P-completeness of some counting problems. Research report 2144, Institut de Recherche en Informatique et en Automatique (December 1993), http://www.lix.polytechnique.fr/~hermann/publications/satcount.ps.gz
Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Information and Computation 125(1), 1–12 (1996)
Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. In: SIAM Monographs on Discrete Mathematics and Applications, vol. 7, SIAM, Philadelphia (2001)
Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 323–332. Springer, Heidelberg (2000): (To appear in Theoretical Computer Science)
Hemaspaandra, L.A., Vollmer, H.: The satanic notations: Counting classes beyond #P and other definitional adventures. SIGACT News, Complexity Theory Column 8 26(1), 2–13 (1995)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the Association for Computing Machinery 44(4), 527–548 (1997)
Jonsson, P., Krokhin, A.: Computational complexity of auditing finite attributes in statistical databases. In: Proceedings Structural Theory of Automata, Semigroups and Universal Algebra, Montreal, Canada (July 2003)
Kleinberg, J., Papadimitriou, C., Raghavan, P.: Auditing Boolean attributes. Journal of Computer and System Science 66(1), 244–253 (2003)
Köbler, J., Schöning, U., Torán, J.: On counting and approximation. Acta Informatica 26(4), 363–379 (1989)
Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. Journal of Computer and System Science 61(2), 302–332 (2000)
Krokhin, A., Jonsson, P.: Recognizing frozen variables in constraint satisfaction problems. Technical Report TR03-062, Electronic Colloquium on Computational Complexity (2003)
Lenzerini, M.: Data integration: a theoretical perspective. In: Proceeding 21st Symposium on Principles of Database Systems (PODS 2002). SIGACT-SIGMOD-SIGART, Madison (Wisconsin, USA), pp. 233–246. ACM Press, New York (2002)
Pippenger, N.: Theories of Computability. Cambridge University Press, Cambridge (1997)
Pöschel, R.: Galois connection for operations and relations. Technical Report MATH-AL-8-2001, Technische Universität Dresden (2001)
Pöschel, R., Kalužnin, L.A.: Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin (1979)
Post, E.L.: The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies 5, 1–122 (1941)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings 10th Symposium on Theory of Computing (STOC 1978), San Diego (California, USA), pp. 216–226 (1978)
Silberstein, C., Brin, S., Motwani, R., Ullman, J.D.: Scalable techniques for mining causal structures. Data Mining and Knowledge Discovery 4(2-3), 163–192 (2000)
Toda, S.: Computational complexity of counting complexity classes. PhD thesis, Tokyo Institute of Technology, Department of Computer Science, Tokyo, Japan (1991)
Toda, S., Watanabe, O.: Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100(1), 205–221 (1992)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)
Widom, J.: Research problems in data warehousing. In: Proceedings 4th International Conference on Information and Knowledge Management (CIKM 1995), Baltimore (Maryland, USA), pp. 25–30. Association for Computing Machinery (1995)
Wrathall, C.: Complete sets and the polynomial-time hierarchy. Theoretical Computer Science 3(1), 23–33 (1976)
Zhuge, Y., Garcia-Molina, H., Hammer, J., Widom, J.: View maintenance in a warehousing environment. In: Carey, M.J., Schneider, D.A. (eds.) Proceedings SIGMOD International Conference on Management of Data, San Jose (California, USA), pp. 316–327. ACM Press, New York (1995)
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Bauland, M., Chapdelaine, P., Creignou, N., Hermann, M., Vollmer, H. (2005). An Algebraic Approach to the Complexity of Generalized Conjunctive Queries. In: Hoos, H.H., Mitchell, D.G. (eds) Theory and Applications of Satisfiability Testing. SAT 2004. Lecture Notes in Computer Science, vol 3542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11527695_3
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DOI: https://doi.org/10.1007/11527695_3
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