The complexity of the equivalence and equation solvability problems over meta-Abelian groups

Abstract

Abstract. We provide polynomial time algorithms for deciding equation solvability and identity checking over groups that are semidirect products of two nite Abelian groups. Our main method is to reduce these problems to the sigma equation solvability and sigma equivalence problems over modules for commutative unital rings.

The equation solvability problem over supernilpotent algebras with Mal’cev term

“…However it is still open whether a complexity dichotomy like over rings holds. In particular nilpotency does not demark the border between problems in P and NP-complete: By [15] the equation solvability over the non-nilpotent group A 4 is in P but its extension by the commutator [·, ·] has an NP-complete equation solvability problem. More general, metaabelian groups [10] and semipattern groups [4] induce equation solvability problems that are in P, while not necessarily being nilpotent.…”

Complexity of term representations of finitary functions

“…x n ] has linear length in n, the corresponding term of A (the one obtained by simply substituting the definition of the commutator for each of its appearance) has length more than 2 n . Unfortunately, we are unable to prove that the term functions t B n cannot be represented by terms in the language of A whose length would be bounded by a polynomial in n. Nevertheless, using [HS12] and under the assumption that P = NP, such terms cannot be produced in polynomial time:…”

“…If P = NP, then there is no algorithm which, given n, produces a term s n in the language of A such that s A n = t B n and which runs in polynomial time in n. Proof. We derive this result from results of [HS12]: the equation solvability problem for A is in P [HS12, Theorem 6]. Furthermore, we will use their reduction of 3colorability to the equation solvability problem for B [HS12, Theorem 13].…”

“…, e k }. In [HS12], the authors produce a term t Γ in the language of B of polynomial length in n and a ∈ A 4 such that t Γ ≈ a has a solution for a = 1 if and only if the graph Γ is 3-colorable. This term is defined by where g (vi,vj ) = y i y −1 j .…”

Notes on extended equation solvability and identity checking for groups

“…We prove the statement by constructing terms in (G, [·, ·]) that allow us to reduce the |G/C G (N )|-graph-coloring problem to Eq(G, [·, ·]) and its complement to Id(G, [·, ·]). Reductions of similar type were already used in [HS12] and [IK17].…”