Abstract
We study the problem of conjunctive query evaluation relative to a class of queries. This problem is formulated here as the relational homomorphism problem relative to a class of structures A, in which each instance must be a pair of structures such that the first structure is an element of A. We present a comprehensive complexity classification of these problems, which strongly links graph-theoretic properties of A to the complexity of the corresponding homomorphism problem. In particular, we define a binary relation on graph classes, which is a preorder, and completely describe the resulting hierarchy given by this relation. This relation is defined in terms of a notion that we call graph deconstruction and that is a variant of the well-known notion of tree decomposition. We then use this hierarchy of graph classes to infer a complexity hierarchy of homomorphism problems that is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized version of quantifier-free, first-order reduction. In doing so, we obtain a significantly refined complexity classification of homomorphism problems as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications. We then present and develop the theory of Ehrenfeucht-Fraïssé-style pebble games, which solve the homomorphism problems where the cores of the structures in A have bounded tree depth. This condition characterizes those classical homomorphism problems decidable in logarithmic space, assuming a hypothesis from parameterized space complexity. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank.
- Serge Abiteboul, Richard Hull, and Victor Vianu. 1995. Foundations of Databases. Addison-Wesley, New York, NY. Google ScholarDigital Library
- Albert Atserias, Andrei A. Bulatov, and Victor Dalmau. 2007. On the power of k-consistency. In 34th International Colloquium on Automata, Languages and Programming (ICALP’07). 279--290. Google ScholarDigital Library
- Albert Atserias and Mark Weyer. 2009. Decidable relationships between consistency notions for constraint satisfaction problems. In 18th Annual Conference of the European Association for Computer Science Logic (CSL’09). 102--116. Google ScholarDigital Library
- Libor Barto and Marcin Kozik. 2009. Constraint satisfaction problems of bounded width. In 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). 595--603. Google ScholarDigital Library
- Achim Blumensath and Bruno Courcelle. 2010. On the monadic second-order transduction hierarchy. Logical Methods in Computer Science 6, 2. Google ScholarCross Ref
- Hans L. Bodlaender. 1998. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209, 1--45. Google ScholarDigital Library
- Andrei A. Bulatov, Andrei A. Krokhin, and Benoit Larose. 2008. Dualities for constraint satisfaction problems. In Complexity of Constraints. 93--124. Google ScholarDigital Library
- Ashok K. Chandra and Philip M. Merlin. 1977. Optimal implementation of conjunctive queries in relational data bases. In 9th Annual ACM Symposium on Theory of Computing (STOC’77). 77--90. Google ScholarDigital Library
- Hubie Chen and Victor Dalmau. 2005. Beyond hypertree width: Decomposition methods without decompositions. In 11th International Conference on Principles and Practice of Constraint Programming. 167--181. Google ScholarDigital Library
- Hubie Chen, Victor Dalmau, and Berit Grußien. 2013. Arc consistency and friends. Journal of Logic and Computation. 23, 1, 87--108. Google ScholarDigital Library
- Hubie Chen and Moritz Müller. 2014. One hierarchy spawns another: Graph deconstructions and the complexity classification of conjunctive queries. Joint 23rd EACSL Computer Science Logic and 29th ACM/IEEE Symposium Logic in Computer Science (CSL-LICS’14), 32:1--32:10. Google ScholarDigital Library
- Hubie Chen and Moritz Müller. 2015. The fine classification of conjunctive queries and parameterized logarithmic space complexity. Transactions on Computation Theory 7, 2, Article No. 7. Google ScholarDigital Library
- Yijia Chen, Jörg Flum, and Martin Grohe. 2003. Bounded nondeterminism and alternation in parameterized complexity theory. In 18th IEEE Conference on Computational Complexity (CCC’03). 13--29.Google ScholarCross Ref
- Yijia Chen and Moritz Müller. 2014. Bounded variable logic, parameterized logarithmic space, and Savitch’s theorem. In 39th International Symposium Mathematical Foundations of Computer Science (MFCS’14). Lecture Notes in Computer Science, Vol. 8634, Springer, Berlin, 183--195.Google ScholarCross Ref
- Victor Dalmau. 2005. Linear datalog and bounded path duality of relational structures. Logical Methods in Computer Science 1, 1. Google ScholarCross Ref
- Victor Dalmau, Phokion G. Kolaitis, and Moshe Y. Vardi. 2002. Constraint satisfaction, bounded treewidth, and finite-variable logics. In Principles and Practice of Constraint Programming (CP). Lecture Notes in Computer Science, Vol. 2470. Springer, Berlin, 310--326. Google ScholarDigital Library
- Anuj Dawar and Yuguo He. 2009. Parameterized complexity classes under logical reductions. In 34th International Symposium Mathematical Foundations of Computer Science (MFCS’09). Lecture Notes in Computer Science, Vol. 5734, Springer, Berlin, 258--269. Google ScholarDigital Library
- Reinhard Diestel. 2012. Graph Theory, 4th Edition. Graduate Texts in Mathematics, Vol. 173. Springer. I--XVIII, 1--436 pages.Google Scholar
- Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. Google ScholarDigital Library
- Heinz-Dieter Ebbinghaus and Jörg Flum. 1995. Finite Model Theory. Springer.Google Scholar
- Michael Elberfeld, Cristoph Stockhusen, and Till Tantau. 2012. On the space complexity of parameterized problems. 7th International Symposium of Parameterized and Exact Computation (IPEC’12). Lecture Notes in Computer Science, Vol. 7535, Springer, Berlin, 206--217. Google ScholarDigital Library
- Jörg Flum and Martin Grohe. 2001. Fixed-parameter tractability, definability and model checking. SIAM Journal on Computing 31, 113--145. Google ScholarDigital Library
- Jörg Flum and Martin Grohe. 2003. Describing parameterized complexity classes. Information and Computation 187, 2, 291--319. Google ScholarDigital Library
- Jörg Flum and Martin Grohe. 2006. Parameterized Complexity Theory. Springer. Google ScholarDigital Library
- Georg Gottlob, Nicola Leone, and Francesco Scarcello. 2001. The complexity of acyclic conjunctive queries. Journal of the ACM 48, 3, 431--498. Google ScholarDigital Library
- Georg Gottlob, Nicola Leone, and Francesco Scarcello. 2002. Hypertree decompositions and tractable queries. Journal of Computer and System Sciences 64, 3, 579--627. Google ScholarDigital Library
- Martin Grohe. 2007. The complexity of homomorphism and constraint satisfaction problems seen from the other side. Journal of the ACM 54, 1. Google ScholarDigital Library
- Martin Grohe, Thomas Schwentick, and Luc Segoufin. 2001. When is the evaluation of conjunctive queries tractable? In 33rd Annual ACM Symposium on Theory of Computing (STOC’01). 657--666. Google ScholarDigital Library
- Neil Immerman. Descriptive Complexity. Springer, 1998.Google Scholar
- Phokion G. Kolaitis and Moshe Y. Vardi. 1995. On the expressive power of Datalog: Tools and a case study. Journal of Computer and System Sciences 51, 110--134. Google ScholarDigital Library
- Phokion G. Kolaitis and Moshe Y. Vardi. 2000a. Conjunctive-query containment and constraint satisfaction. Journal of Computer and System Sciences 61, 302--332. Google ScholarDigital Library
- Phokion G. Kolaitis and Moshe Y. Vardi. 2000b. A game-theoretic approach to constraint satisfaction. In 17th National Conference on AI. 175--181. Google ScholarDigital Library
- Dániel Marx. 2013. Tractable hypergraph properties for constraint satisfaction and conjunctive queries. Journal of the ACM 60, 6, Article No. 42. Google ScholarDigital Library
- Jaroslav Nešetřil and Patrice Ossona de Mendez. Tree depth, subgraph coloring, and homomorphism bounds. European Journal of Combinatorics 27, 6, 1022--1041. Google ScholarDigital Library
- Christos H. Papadimitriou and Mihalis Yannakakis. 1999. On the complexity of database queries. Journal of Computer and System Sciences 58, 3, 407--427. Google ScholarDigital Library
- Neil Robertson and P. D. Seymour. 1983. Graph minors. I. Excluding a forest. Journal of Combinatorial Theory, Series B 35, 1, 39--61. Google ScholarCross Ref
- Neil Robertson and P. D. Seymour. 1986. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory, Series B 41, 1, 92--114, Google ScholarDigital Library
- Nicole Schweikardt, Thomas Schwentick, and Luc Segoufin. 2009. Database theory: Query languages. In Algorithms and Theory of Computation Handbook (2nd ed.), Mikhail J. Atallah and Marina Blanton (Eds.). Vol. 2: Special Topics and Techniques. CRC Press, Boca Raton, FL.Google ScholarDigital Library
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- One Hierarchy Spawns Another: Graph Deconstructions and the Complexity Classification of Conjunctive Queries
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