The Complexity of Conjunctive Queries with Degree 2

TL;DR

This paper extends the understanding of conjunctive query answering complexity to degree 2 hypergraphs, introducing hypergraph dilutions and linking tractability to generalized hypertree width.

Contribution

It introduces hypergraph dilutions as a new tool and characterizes query answering complexity for degree 2 hypergraphs in terms of hypertree width.

Findings

Hypergraph dilutions serve as an alternative to primal graph minors.

An analogue to the Excluded Grid Theorem is established for degree 2 hypergraphs.

Tractability is characterized by generalized hypertree width for degree 2 hypergraphs.

Abstract

We study the tractability of conjunctive query answering for queries with unbounded arity. It is well known that tractability of the problem can be characterised in terms of the queries treewidth under the assumption of bounded arity. We extend this result to cases with unbounded arity but degree 2. To do so, we introduce hypergraph dilutions as an alternative method to primal graph minors for studying substructures of hypergraphs. Using dilutions we observe an analogue to the Excluded Grid Theorem for degree 2 hypergraphs. In consequence, we show that that the tractability of conjunctive query answering can be characterised in terms of generalised hypertree width. A similar characterisation is also shown for the corresponding counting problem. We also generalise our main structural result to arbitrary bounded degree and discuss possible paths towards a characterisation of the bounded degree case.