The Subpower Membership Problem for Finite Algebras with Cube Terms

TL;DR

This paper proves that the Subpower Membership Problem is solvable in polynomial time for finite algebras with cube terms within certain varieties, advancing understanding of algebraic computational complexity.

Contribution

It establishes polynomial-time solvability of SMP for finite algebras with cube terms in residually small varieties and relates SMP to compact representation problems.

Findings

SMP is in P for finite algebras with cube terms in residually small varieties.

SMP, SMP(HS K), and compact representation problems are polynomial-time reducible.

SMP and related problems are in NP for general finite algebras with cube terms.

Abstract

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP we are given a set of tuples in a direct product of algebras from a fixed finite set $\mathcal{K}$ of finite algebras, and are asked whether or not a given tuple belongs to the subalgebra of the direct product generated by a given set. Our main result is that the subpower membership problem SMP($\mathcal{K}$) is in P if $\mathcal{K}$ is a finite set of finite algebras with a cube term, provided $\mathcal{K}$ is contained in a residually small variety. We also prove that for any finite set of finite algebras $\mathcal{K}$ in a variety with a cube term, each one of the problems SMP($\mathcal{K}$), SMP($\mathbb{HS} \mathcal{K}$), and finding compact representations for subpowers in $\mathcal{K}$, is polynomial time reducible to any of the others, and the first two lie in NP.