Answering Conjunctive Queries with Inequalities

In this parer, we study the complexity of answering conjunctive queries (CQ) with inequalities. In particular, we compare the complexity of the query with and without inequalities. The main contribution of our work is a novel combinatorial technique that enables the use of 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. To achieve this, we define a new projection operator that 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 [PODS'97] and give an alternative algorithm based on the color-coding technique [Alon, Yuster and Zwick, PODS'02] 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.