We study constant-delay enumeration for conjunctive queries with negation ($\texttt{CQ}^{\neg}$). Prior work defined \emph{signed-acyclicity}, which characterizes the class of queries where linear preprocessing time is achievable, but little has been known beyond this. We introduce a new hypergraph width measure for $\texttt{CQ}^{\neg}$, the \emph{signed fractional hypertree width} ($\textsf{sfhw}$), defined by requiring that a single variable order simultaneously handle every subset of the negative atoms. We show that $\textsf{sfhw}$ strictly generalizes signed-acyclicity (recovered when $\textsf{sfhw} = 1$) and fractional hypertree width (recovered on queries without negation). Our main algorithmic result is a variable-elimination algorithm that achieves constant-delay enumeration on input $I$ for any full $\texttt{CQ}^{\neg}$ query with preprocessing time $O(|I|^{\textsf{sfhw}})$. We further show that using multiple variable orders can improve this bound, exhibiting an $O(|I|^{3/2})$ algorithm for $k$-cycle queries with at least one negative edge.

翻译：暂无翻译