Tractability Beyond $\beta$-Acyclicity for Conjunctive Queries with Negation

TL;DR

This paper introduces nest-set width, a new hypergraph measure that extends $eta$-acyclicity, enabling tractable evaluation of conjunctive queries with negation and fixed-parameter tractability of propositional satisfiability.

Contribution

It proposes nest-set width as a novel generalization of hypergraph $eta$-acyclicity with proven algorithmic benefits.

Findings

Evaluation of boolean conjunctive queries with negation is tractable for bounded nest-set width.

Propositional satisfiability is fixed-parameter tractable when parameterized by nest-set width.

Nest-set width has desirable properties and algorithmic significance.

Abstract

Numerous fundamental database and reasoning problems are known to be NP-hard in general but tractable on instances where the underlying hypergraph structure is $\beta$-acyclic. Despite the importance of many of these problems, there has been little success in generalizing these results beyond acyclicity. In this paper, we take on this challenge and propose nest-set width, a novel generalization of hypergraph $\beta$-acyclicity. We demonstrate that nest-set width has desirable properties and algorithmic significance. In particular, evaluation of boolean conjunctive queries with negation is tractable for classes with bounded nest-set width. Furthermore, propositional satisfiability is fixed-parameter tractable when parameterized by nest-set width.