A short note on the counting complexity of conjunctive queries

TL;DR

This paper clarifies the counting complexity of conjunctive queries, establishing that non-free-connex queries lack linear-time algorithms and introducing quantified star size as a key complexity measure.

Contribution

It demonstrates that non-free-connex conjunctive queries cannot be counted in linear time and links quantified star size to the runtime lower bounds.

Findings

Non-free-connex queries lack linear-time counting algorithms.

Quantified star size bounds the exponent in counting algorithm runtimes.

Provides a complexity-theoretic characterization of conjunctive query counting.

Abstract

This note closes a minor gap in the literature on the counting complexity of conjunctive queries by showing that queries that are not free-connex do not have a linear time counting algorithm under standard complexity assumptions. More generally, it is shown that the so-called quantified star size is a lower bound for the exponent in the runtime of any counting algorithm for conjunctive queries.