We show that the metaproblem for coset-generating polymorphisms is NP-complete, answering a question of Chen and Larose: given a finite structure, the computational question is whether this structure has a polymorphism of the form $(x,y,z) \mapsto x y^{-1} z$ with respect to some group; such operations are also called coset-generating, or heaps. Furthermore, we introduce a promise version of the metaproblem, parametrised by two polymorphism conditions $Σ_1$ and $Σ_2$ and defined analogously to the promise constraint satisfaction problem. We give sufficient conditions under which the promise metaproblem for $(Σ_1,Σ_2)$ is in P and under which it is NP-hard. In particular, the promise metaproblem is in P if $Σ_1$ states the existence of a Maltsev polymorphism and $Σ_2$ states the existence of an abelian heap polymorphism -- despite the fact that neither the metaproblem for $Σ_1$ nor the metaproblem for $Σ_2$ is known to be in P. We also show that the creation-metaproblem for Maltsev polymorphisms, under the promise that a heap polymorphism exists, is in P if and only if there is a uniform polynomial-time algorithm for CSPs with a heap polymorphism.

翻译：我们证明了陪集生成多态性的元问题是NP完全的，这回答了Chen和Larose提出的一个问题：给定一个有限结构，计算问题是该结构是否具有相对于某个群的形式为$(x,y,z) \mapsto x y^{-1} z$的多态性；此类运算亦称为陪集生成运算或堆。进一步地，我们引入该元问题的承诺版本，该版本由两个多态性条件$Σ_1$和$Σ_2$参数化，其定义方式类比于承诺约束满足问题。我们给出了$(Σ_1,Σ_2)$的承诺元问题属于P的充分条件，以及该问题为NP难的充分条件。特别地，若$Σ_1$断言存在Maltsev多态性且$Σ_2$断言存在阿贝尔堆多态性，则承诺元问题属于P——尽管$Σ_1$的元问题和$Σ_2$的元问题本身均未知是否属于P。我们还证明，在承诺存在堆多态性的前提下，Maltsev多态性的创建-元问题属于P当且仅当存在针对具有堆多态性的CSP的均匀多项式时间算法。