🤖 AI Summary
This paper investigates the direct access problem to the k-th answer in lexicographic order for conjunctive queries with negation (CQ¬) over databases: after polynomial-time preprocessing, arbitrary rank-k queries must be answered in polylogarithmic time. To address this, we systematically extend the characterization of direct-access tractability from positive conjunctive queries to CQ¬, introducing a unified framework based on representable relation circuits. We prove that both β-acyclic and bounded nested set-width classes of negative queries are tractable within this framework, strictly generalizing prior tractability boundaries for positive and negative queries. Experimental evaluation confirms that our approach achieves polynomial preprocessing and polylogarithmic access time, subsuming all previously known tractable classes and extending beyond them.
📝 Abstract
Given a conjunctive query $Q$ and a database $D$, a direct access to the answers of $Q$ over $D$ is the operation of returning, given an index $k$, the $k$-th answer for some order on its answers. While this problem is #P-hard in general with respect to combined complexity, many conjunctive queries have an underlying structure that allows for a direct access to their answers for some lexicographical ordering that takes polylogarithmic time in the size of the database after a polynomial time precomputation. Previous work has precisely characterised the tractable classes and given fine-grained lower bounds on the precomputation time needed depending on the structure of the query. In this paper, we generalise these tractability results to the case of signed conjunctive queries, that is, conjunctive queries that may contain negative atoms. Our technique is based on a class of circuits that can represent relational data. We first show that this class supports tractable direct access after a polynomial time preprocessing. We then give bounds on the size of the circuit needed to represent the answer set of signed conjunctive queries depending on their structure. Both results combined together allow us to prove the tractability of direct access for a large class of conjunctive queries. On the one hand, we recover the known tractable classes from the literature in the case of positive conjunctive queries. On the other hand, we generalise and unify known tractability results about negative conjunctive queries -- that is, queries having only negated atoms. In particular, we show that the class of $eta$-acyclic negative conjunctive queries and the class of bounded nest set width negative conjunctive queries admit tractable direct access.