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    One hierarchy spawns another: graph deconstructions and the complexity classification of conjunctive queries

    Chen, Hubie and Mueller, M. (2017) One hierarchy spawns another: graph deconstructions and the complexity classification of conjunctive queries. ACM Transactions on Computational Logic 18 (4), ISSN 1529-3785.


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    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.


    Item Type: Article
    Additional Information: © ACM, 2017. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published at the link above.
    School: Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
    Depositing User: Hubie Chen
    Date Deposited: 11 Apr 2018 17:17
    Last Modified: 09 Aug 2023 12:43


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