No-Rainbow Problem and the Surjective Constraint Satisfaction Problem

The Surjective Constraint Satisfaction Problem (SCSP) is the problem of deciding whether there exists a surjective assignment to a set of variables subject to some specified constraints. In this paper we show that one of the most popular variants of the SCSP, called No-Rainbow Problem, is NP-Hard. Additionally, we disprove the conjecture saying that the SCSP over a constraint language $\Gamma$ is equivalent to the CSP over the same language with constants. Our counter example also shows that the complexity of the SCSP cannot be described in terms of polymorphisms of the constraint language.

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