On Tractability and Hypertree Width

We investigate in this paper the notion of hypertree width as a parameter for bounding the complexity of CSPs, especially those whose constraints can be represented compactly, such as SAT problems. We first identify a simple condition which is necessary for hypertree width to provide better complexity bounds than treewidth. We then observe that SAT problems do not satisfy this condition and, hence, hypertree width cannot directly provide tighter bounds than those obtained by treewidth on these problems. We next identify a simple class of SAT problems which may contain tractable subsets, yet neither hypertree width nor treewidth bounds can recognize such tractability. Hence our final contribution is a technique for introducing the auxiliary variables into the problems that allows us to recognize the tractability of this class of problems.