The complexity of finding coset-generating polymorphisms and the promise metaproblem

TL;DR

This work studies the metaproblem for coset-generating polymorphisms (heaps) within finite structures, proving that Meta for the coset-generating condition is NP-complete and introducing promise metaproblems that generalize the framework. It establishes a nuanced connection between promise metaproblems and uniform CSPs, showing tractability in cases where Σ1 is idempotent and Σ2 is a linear strong Maltsev condition, and specifically that PMeta(Σ_heap,Σ_Maltsev) sits in P. The authors prove that a uniform algorithm for CSPs with coset-generating polymorphisms exists if and only if a corresponding promise creation-metaproblem is in P, and they extend hardness results to promise metaproblems as well. They provide a concrete NP-hardness reduction from a graph decomposition problem using dihedral groups, and discuss tractable subcases such as abelian heaps, highlighting both the depth and boundaries of tractability in this domain. The results clarify the landscape of uniform algorithms for coset CSPs and outline open questions about abelian heap recognition and related polymorphism conditions.

Abstract

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.

Figures (1)

• Figure 1: For each variable $X_k$ ($k \in \{1, \dots, n\}$) we have one of these subgraphs in the proof of Lemma \ref{['lem:graphNPcomplete']}. Here the vertices $u_1, \dots, u_q$ correspond to those vertices $v_{i,j}$ where the literal $L_{i,j}$ is the variable $X_k$. Note that $q \geqslant 3$.

Theorems & Definitions (19)

• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 4: Theorem 6.2 in MetaChenLarose
• Proposition 6
• Proposition 7
• Proposition 8
• Lemma 9
• Lemma 10
• Corollary 11