On Containment of Conjunctive Queries with Arithmetic Comparisons

We study the following problem: how to test if Q2 is contained in Q1, where Q1 and Q2 are conjunctive queries with arithmetic comparisons? This problem is fundamental in a large variety of database applications. Existing algorithms rst normalize the queries, then test a logical implication using multiple containment mappings from Q1 to Q2. We are interested in cases where the containment can be tested more eÆciently. This work aims to (a) reduce the problem complexity from P 2 -completeness to NP-completeness in these cases; (b) utilize the advantages of the homomorphism property (i.e., the containment test is based on a single containment mapping) in applications such as those of answering queries using views; and (c) observing that many real queries have the homomorphism property. The following are our results. (1) We show several cases where the normalization step is not needed, thus reducing the size of the queries and the number of containment mappings. (2) We develop an algorithm for checking various syntactic conditions on queries, under which the homomorphism property holds. (3) We further reduce the conditions of these classes using practical domain knowledge that is easily obtainable. (4) We conducted experiments on real queries, and show that most of the queries pass this test.