Armin Weiß

Manuel Bodirsky, Armin Weiß

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.

Eric Samperton, Armin Weiß

Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $Γ$, count the number of homomorphisms $Γ\to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $Γ$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $Γ$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $Γ= π_1(M^3)$ for some input 3-manifold triangulation $M^3$ with $|H^2(M,Z(G)|$ bounded above, then there is a polynomial time algorithm to compute the number of homomorphisms from $Γ$ to $G$. This algorithm is explained in part by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to solve the necessary group cohomological obstruction problems efficiently using the given triangulation. A similar polynomial time algorithm for counting maps to finite, class 2 nilpotent $G$ exists when $Γ$ is itself a finite group encoded via a multiplication table, provided that $|H^2(Γ,Z(G))|$ is similarly bounded from above.