寻找共集生成多态的复杂性及承诺元问题
The complexity of finding coset-generating polymorphisms and the promise metaproblem
AI总结：
研究证明寻找共集生成多态的元问题为NP完全，提出承诺版本的元问题，并探讨其在P和NP难的条件，涉及Maltsev多态和交换堆多态的存在性。
摘要：
我们证明寻找共集生成多态的元问题是NP完全，回答了Chen和Larose的问题：给定有限结构，判断是否存在某种群下的多态形式(x,y,z)↦xy⁻¹z。此外，我们引入了由两个多态条件Σ₁和Σ₂参数化的承诺元问题，给出其在P和NP难的充分条件。特别是，当Σ₁要求存在Maltsev多态，Σ₂要求存在交换堆多态时，承诺元问题在P中，尽管单独的Σ₁和Σ₂元问题未知是否在P中。我们还证明在存在堆多态的承诺下，Maltsev多态的创建元问题在P中当且仅当存在统一的多项式时间算法解决具有堆多态的CSP。
英文摘要：
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.
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