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The Complexity of Finding Coset-Generating Polymorphisms and the Promise Metaproblem

Authors Manuel Bodirsky , Armin Weiß

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LIPIcs.ICALP.2026.169.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • constraint satisfaction problem
  • coset-generating polymorphisms
  • metaproblem
  • heap
  • abelian heap
  • uniform polynomial-time algorithm
  • NP-hardness

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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) ↦ 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 Σ₁ and Σ₂ and defined analogously to the promise constraint satisfaction problem. We give sufficient conditions under which the promise metaproblem for (Σ₁,Σ₂) is in 𝖯 and under which it is NP-hard. In particular, the promise metaproblem is in 𝖯 if Σ₁ states the existence of a Maltsev polymorphism and Σ₂ states the existence of an abelian heap polymorphism - despite the fact that neither the metaproblem for Σ₁ nor the metaproblem for Σ₂ 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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Manuel Bodirsky and Armin Weiß. The Complexity of Finding Coset-Generating Polymorphisms and the Promise Metaproblem. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 169:1-169:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.169

Author Details

Manuel Bodirsky
  • Institut für Algebra, TU Dresden, Germany
Armin Weiß
  • FMI, University of Stuttgart, Germany

Funding

  • Bodirsky, Manuel: Received funding from the ERC (Grant Agreement no. 101071674, POCOCOP). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency.
  • Weiß, Armin: Supported by the German Research Foundation (DFG) grant WE 6835/1-2.

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