Complexity of Conjunctive Queries
For the study of the computational complexity of evaluating conjunctive queries, two problems have to be distinguished. The first is the problem of evaluating a conjunctive query on a relational database where both the query and the database are considered part of the input. The complexity of this problem is usually referred to as combined complexity, while the complexity of the problem of evaluating a query on a relational database, where the query is assumed fixed, is called data complexity.
Conjunctive queries are NP-complete with respect to combined complexity, while the data complexity of conjunctive queries is very low, in the parallel complexity class AC0, which is contained in LOGSPACE and thus in polynomial time. The NP-hardness of conjunctive queries may appear surprising, since relational algebra and SQL strictly subsume the conjunctive queries and are thus at least as hard (in fact, relational algebra is PSPACE-complete with respect to combined complexity and is therefore even harder under widely-held complexity-theoretic assumptions). However, in the usual application scenario, databases are large, while queries are very small, and the data complexity model may be appropriate for studying and describing their difficulty.
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