One Hierarchy Spawns Another: Graph Deconstructions and the Complexity Classification of Conjunctive Queries

ABSTRACT. We study the problem of conjunctive query evaluation over classes of queries; this problem is formulated here as the relational homomorphism problem over a class of structures A, wherein 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 and completely describe the resulting hierarchy given by this relation. This binary relation is defined in terms of a notion which we call graph deconstruction and which is a variant of the well-known notion of tree decomposition. We then use this graph hierarchy to infer a complexity hierarchy of homomorphism problems which is comprehensive up to a computationally very weak notion of reduction, namely, a parameterized version of quantifier-free reductions. 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-Fra\"{i}ss\'{e}-style pebble games which solve the homomorphism problems where the cores of the structures in A have bounded tree depth; as a fruit of this development, we characterize the homomorphism problems solvable in logarithmic space. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank.