The Dichotomy for Conservative Constraint Satisfaction is Polynomially Decidable

Given a fixed constraint language Γ , the conservative CSP over Γ (denoted by c-CSP(Γ )) is a variant of CSP(Γ ) where the domain of each variable can be restricted arbitrarily. In [5] a dichotomy has been proven for conservative CSP: for every fixed language Γ , c-CSP(Γ ) is either in P or NP-complete. However, the characterization of conservatively tractable languages is of algebraic nature and the recognition algorithm provided in [5] is super-exponential in the domain size. The main contribution of this paper is a polynomial-time algorithm that, given a constraint language Γ as input, decides if c-CSP(Γ ) is tractable. In addition, if Γ is proven tractable the algorithm also outputs its coloured graph, which contains valuable information on the structure of Γ .