Answering Conjunctive Queries with Inequalities

TL;DR

This paper introduces a new combinatorial technique for efficiently answering conjunctive queries with inequalities by leveraging existing query plans and analyzing graph structures, enabling polynomial-time evaluation in certain cases.

Contribution

The work presents a novel projection operator, generalizes existing algorithms, and analyzes query and inequality graph structures to identify classes with polynomial-time evaluation.

Findings

A new projection operator reduces inequality evaluation complexity.

An alternative algorithm based on color-coding evaluates queries with inequalities.

Certain query and inequality graph structures allow polynomial-time evaluation.

Abstract

In this paper, we study the complexity of answering conjunctive queries (CQ) with inequalities). In particular, we are interested in comparing the complexity of the query with and without inequalities. The main contribution of our work is a novel combinatorial technique that enables us to use any Select-Project-Join query plan for a given CQ without inequalities in answering the CQ with inequalities, with an additional factor in running time that only depends on the query. The key idea is to define a new projection operator, which keeps a small representation (independent of the size of the database) of the set of input tuples that map to each tuple in the output of the projection; this representation is used to evaluate all the inequalities in the query. Second, we generalize a result by Papadimitriou-Yannakakis [17] and give an alternative algorithm based on the color-coding technique [4] to evaluate a CQ with inequalities by using an algorithm for the CQ without inequalities. Third, we investigate the structure of the query graph, inequality graph, and the augmented query graph with inequalities, and show that even if the query and the inequality graphs have bounded treewidth, the augmented graph not only can have an unbounded treewidth but can also be NP-hard to evaluate. Further, we illustrate classes of queries and inequalities where the augmented graphs have unbounded treewidth, but the CQ with inequalities can be evaluated in poly-time. Finally, we give necessary properties and sufficient properties that allow a class of CQs to have poly-time combined complexity with respect to any inequality pattern. We also illustrate classes of queries where our query-plan-based technique outperforms the alternative approaches discussed in the paper.