Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations

TL;DR

This paper classifies the complexity of approximately counting answers to small conjunctive queries with disequalities and negations, establishing tractability conditions based on width measures and the randomised Exponential Time Hypothesis.

Contribution

It provides a comprehensive classification of when efficient approximation schemes exist for counting query answers with disequalities and negations, extending previous results to more general query classes.

Findings

FPTRAS exists if treewidth or adaptive width is bounded under rETH

No FPRAS unless NP=RP, even for treewidth 1

FPRAS exists for queries without disequalities or negations with bounded fractional hypertreewidth

Abstract

We study the complexity of approximating the number of answers to a small query $\varphi$ in a large database $\mathcal{D}$. We establish an exhaustive classification into tractable and intractable cases if $\varphi$ is a conjunctive query with disequalities and negations: $\bullet$ If there is a constant bound on the arity of $\varphi$, and if the randomised Exponential Time Hypothesis (rETH) holds, then the problem has a fixed-parameter tractable approximation scheme (FPTRAS) if and only if the treewidth of $\varphi$ is bounded. $\bullet$ If the arity is unbounded and we allow disequalities only, then the problem has an FPTRAS if and only if the adaptive width of $\varphi$ (a width measure strictly more general than treewidth) is bounded; the lower bound relies on the rETH as well. Additionally we show that our results cannot be strengthened to achieve a fully polynomial randomised approximation scheme (FPRAS): We observe that, unless $\mathrm{NP} =\mathrm{RP}$, there is no FPRAS even if the treewidth (and the adaptive width) is $1$. However, if there are neither disequalities nor negations, we prove the existence of an FPRAS for queries of bounded fractional hypertreewidth, strictly generalising the recently established FPRAS for conjunctive queries with bounded hypertreewidth due to Arenas, Croquevielle, Jayaram and Riveros (STOC 2021).