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 conceptsÃ‚Â·
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 2002, 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 by
Gottlob et al. in 2007, 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
cle~r 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 eV,en 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.