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.

中文翻译：

---

无彩虹问题和满射约束满足问题

满射约束满足问题 (SCSP) 是决定是否存在对受某些特定约束的一组变量进行满射赋值的问题。在本文中，我们展示了 SCSP 最流行的变体之一，称为 No-Rainbow Problem，是 NP-Hard。此外，我们反驳了以下猜想：约束语言 $\Gamma$ 上的 SCSP 等价于具有常量的相同语言上的 CSP。我们的反例还表明 SCSP 的复杂性不能用约束语言的多态性来描述。