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

TL;DR

This paper provides a detailed complexity classification of conjunctive query evaluation by linking graph properties to homomorphism problem complexity, introducing a hierarchy based on graph deconstruction and pebble games.

Contribution

It introduces a hierarchy of graph classes via graph deconstruction, refining the complexity classification of homomorphism problems and unifying existing results.

Findings

Hierarchy of graph classes defined by graph deconstruction.

Refined complexity classification of homomorphism problems.

Characterization of model checking for bounded quantifier rank.

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, 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, which is a preorder, and completely describe the resulting hierarchy given by this relation. This 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 hierarchy of graph classes 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 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-Fraisse-style pebble games which solve the homomorphism problems where the cores of the structures in A have bounded tree depth. Finally, we use our framework to classify the complexity of model checking existential sentences having bounded quantifier rank.