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).

中文翻译：

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带有不等式和否定的联合查询的近似计数答案

我们研究了在大型数据库$ \ mathcal {D} $中估算小型查询$ \ varphi $的答案数量的复杂度。如果$ \ varphi $是带有不等式和否定的合取查询，我们将穷举性分类为难处理的情况：$ \ bullet $如果$ \ varphi $的对偶数有一个常数边界，并且随机指数时间假说（rETH）成立，那么当且仅当$ \ varphi $的树宽是有界的时，问题才具有固定参数易处理的近似方案（FPTRAS）。$ \ bullet $如果无穷大且我们仅允许不等式，则当且仅当$ \ varphi $的自适应宽度（严格比树宽更严格的宽度度量）有界时，问题才具有FPTRAS；下限也依赖于rETH。此外，我们证明了无法增强我们的结果以实现完全多项式随机逼近方案（FPRAS）：我们观察到，除非$ \ mathrm {NP} = \ mathrm {RP} $，否则即使树宽（和（自适应宽度）为$ 1 $。但是，如果既没有不等式也没有否定性，我们证明存在FPRAS用于有限分数超树宽查询，严格地归纳了最近建立的FPRAS用于由于Arenas，Croquevielle，Jayaram和Riveros（STOC 2021）而导致的有界超树宽的联合查询。