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：我们证明同余生成多态性的元问题是NP完全的，回答了Chen与Larose提出的问题：给定一个有限结构，计算该结构是否具有关于某个群的形如$(x,y,z) \mapsto x y^{-1} z$的多态性；此类操作也被称为同余生成或堆操作。进一步地，我们引入元问题的承诺版本，该版本由两个多态性条件$\Sigma_1$和$\Sigma_2$参数化，其定义方式类似于承诺约束满足问题。我们给出充分条件，使得$(\Sigma_1,\Sigma_2)$的承诺元问题属于P类或为NP难问题。特别地，当$\Sigma_1$声明Maltsev多态性存在而$\Sigma_2$声明交换堆多态性存在时，承诺元问题属于P类——尽管$\Sigma_1$和$\Sigma_2$各自的元问题均未知属于P类。我们还证明，在承诺堆多态性存在的前提下，Maltsev多态性的创建元问题属于P类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。