Tractable Decision for a Constraint Language Implies Tractable Search

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A constraint satisfaction problem (CSP) instance has a set of variables, each of which can take values in some domain. It also has a set of constraints, each of which restricts the variables in its scope to take values limited by its constraint relation.

The language of a constraint satisfaction problem instance is the set of different constraint relations used in its specification. For a given set of relations Γ over some domain we define the problem CSP (Γ) to the set of CSP instances whose language is contained in Γ.

The decision problem for a set of CSP instances is, given an instance in the class, to decide whether a solution exists. The search problem is to find such a solution. Here we address the connection between the tractability of the decision and search problems. We prove that given a constraint language Γ over a finite domain for which the decision problem for CSP (Γ) is tractable, the search problem is always tractable.

We define a surjective language over a finite domain in a standard way. We also show that we can determine in polynomial time whether an instance over a surjective language with a tractable decision problem has fewer than k solutions, and that we can generate all of its solutions with polynomial delay.

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