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In this paper, we study the complexity of answering conjunctive queries (CQ)with inequalities). In particular, we are interested in comparing thecomplexity of the query with and without inequalities. The main contribution ofour work is a novel combinatorial technique that enables us to use anySelect-Project-Join query plan for a given CQ without inequalities in answeringthe CQ with inequalities, with an additional factor in running time that onlydepends 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) ofthe 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 analternative algorithm based on the color-coding technique [4] to evaluate a CQwith inequalities by using an algorithm for the CQ without inequalities. Third,we investigate the structure of the query graph, inequality graph, and theaugmented query graph with inequalities, and show that even if the query andthe inequality graphs have bounded treewidth, the augmented graph not only canhave an unbounded treewidth but can also be NP-hard to evaluate. Further, weillustrate classes of queries and inequalities where the augmented graphs haveunbounded treewidth, but the CQ with inequalities can be evaluated inpoly-time. Finally, we give necessary properties and sufficient properties thatallow a class of CQs to have poly-time combined complexity with respect to anyinequality pattern. We also illustrate classes of queries where ourquery-plan-based technique outperforms the alternative approaches discussed inthe paper.

Answering Conjunctive Queries with Inequalities