The constraint satisfaction problem (CSP) on a finite relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is a assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic framework for proving hardness results on surjective CSPs; essentially, this framework computes global gadgetry that permits one to present a reduction from a classical CSP to a surjective CSP. We show how to derive a number of hardness results for surjective CSP in this framework, including the hardness of the disconnected cut problem, of the no-rainbow 3-coloring problem, and of the surjective CSP on all 2-element structures known to be intractable (in this setting). Our framework thus allows us to unify these hardness results, and reveal common structure among them; we believe that our hardness proof for the disconnected cut problem is more succinct than the original. In our view, the framework also makes very transparent a way in which classical CSPs can be reduced to surjective CSPs.

翻译：约束满足问题(CSP)在有限关系结构B上是指：给定一组变量约束（其中关系来自B），判定是否存在一个满足所有约束的变量赋值；满射CSP是其中的变体，要求判定是否存在一个到B论域上的满射满足赋值。我们提出了一个用于证明满射CSP难解性结果的代数框架；本质上，该框架计算全局构建方法，使得能够从经典CSP到满射CSP进行归约。我们展示了如何在此框架中推导满射CSP的多个难解性结果，包括不连通割问题、无彩虹3-染色问题以及在所有已知难解的2元结构上的满射CSP（在该设定下）的难解性。因此，我们的框架能够统一这些难解性结果，并揭示它们之间的共同结构；我们相信对不连通割问题的难解性证明比原始证明更为简洁。在我们看来，该框架还非常清晰地展示了经典CSP如何被归约到满射CSP的方法。