Hypertree Decompositions and Tractable Queries

摘要

Several important decision problems on conjunctive queries (CQs) are NP-complete in general but become tractable, and actually highly parallelizable, if restricted to acyclic or nearly acyclic queries. Examples are the evaluation of Boolean CQs and query containment. These problems were shown tractable for conjunctive queries of bounded treewidth (Ch. Chekuri and A. Rajaraman, Theoret. Comput. Sci.239 (2000), 211–229), and of bounded degree of cyclicity (M. Gyssens et al., Artif. Intell.66 (1994), 57–89; M. Gyssens and J. Paredaens, in “Advances in Database Theory, Vol. 2, pp. 85–122, Plenum Press, New York, 1984). The so far most general concept of nearly acyclic queries was the notion of queries of bounded query-width introduced by Chekuri and Rajaraman (2000). While CQs of bounded query-width are tractable, it remained unclear whether such queries are efficiently recognizable. Chekuri and Rajaraman (2000) stated as an open problem whether for each constant k it can be determined in polynomial time if a query has query-width at most k. We give a negative answer by proving the NP-completeness of this problem (specifically, for k=4). In order to circumvent this difficulty, we introduce the new concept of hypertree decomposition of a query and the corresponding notion of hypertree-width. We prove: (a) for each k, the class of queries with query-width bounded by k is properly contained in the class of queries whose hypertree-width is bounded by k; (b) unlike query-width, constant hypertree-width is efficiently recognizable; and (c) Boolean queries of bounded hypertree-width can be efficiently evaluated.