Polynomial-time tractability for finite 2-step nilpotent Mal'cev algebras

Determine whether the subpower membership problem belongs to the complexity class P for every finite 2-step nilpotent Mal'cev algebra of finite type.

Background

Finite supernilpotent Mal'cev algebras have polynomial-time solvable subpower membership problems, but finite nilpotent Mal'cev algebras need not be supernilpotent. The paper discusses a 12-element 2-step nilpotent loop that is not supernilpotent, while its subpower membership problem is nevertheless tractable.

The paper proves polynomial-time tractability for a substantial class of 2-step nilpotent Mal'cev algebras, including those of squarefree order and central extensions satisfying specified structural conditions. It leaves open whether every finite 2-step nilpotent Mal'cev algebra of finite type has this property.

References

Is $\textup{SMP}(A) \in \textup{P}$ for every finite $2$-step nilpotent Mal'cev algebra of finite type?

— Structure and Complexity of 2-Nilpotent Mal'cev Algebras  (2608.18917 - Wynne, 19 Aug 2026) in Question, Introduction

It remains open whether every finite $2$-nilpotent Mal'cev algebra of finite type has tractable subpower membership problem.

— Structure and Complexity of 2-Nilpotent Mal'cev Algebras  (2608.18917 - Wynne, 19 Aug 2026) in Conclusion of the proof of Corollary \ref{thm:SMPcorollary}, Section 3