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Vegaswikian (talk | contribs) Disambiguated: tractable → tractable problem; Unlinked: Primal graph (2); Help needed: Primal graph |
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By definition, a decomposition method produces a binary acyclic problem; such problems can be solved in time polynomial in its size. As a result, the original problem can be solved by first translating it and then solving the resulting problem; however, this algorithm is polynomial-time only if the decomposition does not increase size superpolynomially. The ''width'' of a decomposition method is a measure of the size of problem it produced. Originally, the width was defined as the maximal cardinality of the sets of original variables; one method, the hypertree decomposition, uses a different measure. Either way, the width of a decomposition is defined so that decompositions of size bounded by a constant do not produce excessively large problems. Instances having a decomposition of fixed width can be translated by decomposition into instances of size bounded by a polynomial in the size of the original instance.
The width of a problem is the width of its minimal-width decomposition. While decompositions of fixed width can be used to efficiently solve a problem, a bound on the width of instances does necessarily produce a [[tractable problem|tractable]] [[structural restriction]]. Indeed, a fixed width problem has a decomposition of fixed width, but finding it may not be polynomial. In order for a problem of fixed width being efficiently solved by decomposition, one of its decompositions of low width has to be found efficiently. For this reason, decomposition methods and their associated width are defined in such a way not only solving the problem given a fixed-width decomposition of it is polynomial-time, but also finding a fixed width decomposition of a fixed-width problem is polynomial-time.
==Decomposition methods==
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Decomposition methods create a problem that is easy to solve from an arbitrary one. Each variable of this new problem is associated to a set of original variables; its ___domain contains tuples of values for the variables in the associated set; in particular, these are the tuples that satisfy a set of constraints over these variables. The constraints of the new problem bounds the values of two new variables to have as values two tuples that agree on the shared original variables. Three further conditions ensure that the new problem is equivalent to the old one and can be solved efficiently.
In order for the new problem to be solvable efficiently, the [[primal graph]]{{dn|date=April 2013}} of the new problem is required to be acyclic. In other words, viewing the variables as vertices and the (binary) constraints as edges, the resulting graph is required to be a [[Tree (graph theory)|tree]] or a [[Forest (graph theory)|forest]].
In order for the new problem to be equivalent to the old one, each original constraint is enforced as part of the definition of the ___domain of at least one new variables. This requires that, for each constraint of the old problem, there exists a variable of the new problem such that its associated set of original variables include the scope of the constraint, and all tuples in its ___domain satisfy the constraint.
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===Biconnected components===
The biconnected decomposition of an arbitrary constraint satisfaction problem is the biconnected decomposition of its
===Tree decomposition===
A tree decomposition of an arbitrary constraint satisfaction problem is a tree decomposition of its
===Cycle hypercutset===
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