A Unified Theory of Structural Tractability for Constraint Satisfaction and Spread Cut Decomposition

In this paper we introduce a generic form of structural decomposition for the constraint satisfaction problem, which we call a guarded decomposition. We show that many existing decomposition methods can be characterized in terms of finding guarded decompositions satisfying certain specified additional conditions. Using the guarded decomposition framework we are also able to define a new form of decomposition, which we call a spread cut. We show that discovery of width k spread-cut decompositions is tractable for each k, and that the spread cut decomposition strongly generalize all existing decompositions except hypertrees. Finally we exhibit a family of hypergraphs Hn, for n = 1, 2, 3 . . ., where the width of the best hypertree decomposition of each Hn is at least 3n, but the width of the best spreadcut decomposition is at most 2n.