Understanding Tractable Decompositions for Constraint Satisfaction

Constraint satisfaction problems (CSPs) are NP-complete in general, therefore it is important to identify tractable subclasses. A possible way to find such subclasses is to associate a hypergraph to the problem and impose restrictions on its structure. In this thesis we follow this direction. Among such structural properties, particularly important is acyclicity: it is well known that CSPs whose associated hypergraph is acyclic can be solved efficiently. In the last decade, many structural decompositions have been proposed. These concepts can be seen as generalizations of hypergraph acyclicity. The interesting decomposition concepts are those which both enable the problems in the defined subclass to be solved in polynomial time and the associated hypergraphs to be recognized efficiently. Hypertree decompositions, introduced by Gottlob et al. in [43], fall in this category and additionally, for a long time, this class was the most general concept known to have both of these desirable properties. We study further generalizations of this concept. It was shown recently ([45]) that the recognition problem for the most straightforward generalization, for the so called generalized hypertree decompositions, is NP-hard. Understanding the deep reasons for this intractability result enabled us to define new decompositions with tractable recognition algorithms. We not only introduce a new decomposition concept, but also a methodology to define such decompositions using subedges of the hypergraph. In this way we get a very clear picture of tractable decompositions. As an application of our method, we construct a new decomposition concept, called component hypertree decomposition, which is tractable and strictly more general than all other known tractable methods, including the recently introduced spread cut decomposition. We also define an even more general concept, which also generalizes the spread cut decompositions, according to their new definitions. We analyze various properties of generalized hypertree decompositions and study the parallel complexity of the recognition algorithms for the known tractable decomposition methods. Understanding their similarities and their relation to generalized hypertree decomposition, we gave upper bounds for the parallel complexity of their recognition.