Approximately Counting Answers to Conjunctive Queries with Disequalities and Negations
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 fixedparameter 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).
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.12468
 arXiv:
 arXiv:2103.12468
 Bibcode:
 2021arXiv210312468F
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Computer Science  Databases
 EPrint:
 An extended abstract of this work is accepted for publication at PODS22. 30 pages, 1 figure