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

Hubie Chen (Universidad del País Vasco and IKERBASQUE)We study the classical problem of conjunctive query evaluation, here restricted according to the set of permissible queries. In this work, this problem is formulated as the relational homomorphism problem over a set 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 form of quantifier-free reductions. We obtain a significantly refined complexity classification of left-hand side restricted homomorphism problems, as well as a unifying, modular, and conceptually clean treatment of existing complexity classifications, such as the classifications by Grohe-Schwentick-Segoufin (STOC 2001) and Grohe (FOCS 2003, JACM 2007).

In this talk, we will also briefly discuss parameterized complexity classes that we introduced/studied which capture some of the complexity degrees identified by our classification.

This talk is based on joint work with Moritz Müller that appeared in PODS’13 and CSL-LICS’14.