Decomposition method (constraint satisfaction): Difference between revisions

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==Overview==

Decomposition methods translate a problem into a new one that is easier to solve. The new problem only contains [[binary constraint]]s; their scopes form ana [[directed acyclic graph]]. The variables of the new problem represent each a set of variables of the original one. These sets are not necessarily disjoint, but they cover the set of the original variables. The translation finds all partial solutions relative to each set of variables. The problem that results from the translation represents the interactions between these local solutions.

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.