Boolean Tensor Decomposition for Conjunctive Queries with Negation

TL;DR

This paper introduces an algorithm for efficiently answering conjunctive queries with negation on bounded-degree relations, using Boolean tensor decomposition and query rewriting techniques, with complexity depending on query structure.

Contribution

It presents a novel approach combining query rewriting, NAE predicates, and tensor decomposition to handle negation in conjunctive queries with bounded-degree relations.

Findings

Data complexity matches the best known algorithms for positive subqueries.

Query complexity is exponential in negated join variables but polynomial for certain classes.

The proposed method can be derandomized efficiently using probabilistic techniques.

Abstract

We propose an algorithm for answering conjunctive queries with negation, where the negated relations have bounded degree. Its data complexity matches that of the best known algorithms for the positive subquery of the input query and is expressed in terms of the fractional hypertree width and the submodular width. The query complexity depends on the structure of the negated subquery; in general it is exponential in the number of join variables occurring in negated relations yet it becomes polynomial for several classes of queries. This algorithm relies on several contributions. We show how to rewrite queries with negation on bounded-degree relations into equivalent conjunctive queries with not-all-equal (NAE) predicates, which are a multi-dimensional analog of disequality (not-equal). We then generalize the known color-coding technique to conjunctions of NAE predicates and explain it via a Boolean tensor decomposition of conjunctions of NAE predicates. This decomposition can be achieved via a probabilistic construction that can be derandomized efficiently.