Conjunctive Queries with Negation: Beyond Signed-Acyclicity

2026-10-02

AI Summary

This study addresses the limitation of existing constant-delay enumeration theory for conjunctive queries with negation, which is confined to signed acyclicity and thus struggles with broader query classes. To overcome this, the work proposes symbolic fractional hypertree width (sfhw) as a novel hypergraph width measure that strictly generalizes both signed acyclicity and fractional hypertree width. Based on this measure, an enumeration algorithm leveraging variable elimination and multi-variable ordering optimization is designed. By breaking through prior theoretical constraints, the approach achieves constant-delay enumeration with O(|I|^sfhw) preprocessing time and optimizes the complexity of k-cycle queries to O(|I|^{3/2}), significantly extending the applicability boundaries of efficient enumeration.

Abstract

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.

Problem

• conjunctive queries with negation
• constant-delay enumeration
• signed-acyclicity
• fractional hypertree width

Innovation

• Conjunctive Queries with Negation
• Signed Fractional Hypertree Width
• Constant-Delay Enumeration
• Variable Elimination
• Hypergraph Width Measure