The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space Complexity

TL;DR

This paper provides a detailed classification of conjunctive query evaluation complexity within parameterized logarithmic space, revealing three degrees of complexity based on structural properties like pathwidth and tree depth.

Contribution

It introduces a fine-grained classification of conjunctive query sets up to parameterized logarithmic space reduction, extending previous fixed-parameter tractability results.

Findings

Three complexity degrees within bounded treewidth regime.

Bounded pathwidth and tree depth determine the complexity degree.

Logarithmic space machine characterizations for higher degrees.

Abstract

We perform a fundamental investigation of the complexity of conjunctive query evaluation from the perspective of parameterized complexity. We classify sets of boolean conjunctive queries according to the complexity of this problem. Previous work showed that a set of conjunctive queries is fixed-parameter tractable precisely when the set is equivalent to a set of queries having bounded treewidth. We present a fine classification of query sets up to parameterized logarithmic space reduction. We show that, in the bounded treewidth regime, there are three complexity degrees and that the properties that determine the degree of a query set are bounded pathwidth and bounded tree depth. We also engage in a study of the two higher degrees via logarithmic space machine characterizations and complete problems. Our work yields a significantly richer perspective on the complexity of conjunctive queries and, at the same time, suggests new avenues of research in parameterized complexity.