Hypertree Decompositions and Tractable Queries
  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 and of bounded degree of cyclicity. The so far most general concept of nearly acyclic queries was the notion of queries of bounded query-width introduced by Chekuri and Rajaraman (1997). While CQs of bounded query width are tractable, it remained unclear whether such queries are efficiently recognizable. Chekuri and Rajaraman stated as an open problem whether for each constant k it can be determined in polynomial time if a query has query width less than or equal to k. We give a negative answer by proving this problem NP-complete (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; (c) Boolean queries of constant hypertree width can be efficiently evaluated. 
G. Gottlob, N. Leone, F. Scarcello