The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.
The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge 0$ with $g_1^{x_1}\cdots g_n^{x_n}=g$. We study the knapsack problem for wreath products $G\wr H$ of groups $G$ and $H$. Our main result is a characterization of those wreath products $G\wr H$ for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors $G$ and $H$. To this end, we introduc…
Recently knapsack problems have been generalized from the integers to arbitrary finitely generated groups. The knapsack problem for a finitely generated group $G$ is the following decision problem: given a tuple $(g, g_1, \ldots, g_k)$ of elements of $G$, are there natural numbers $n_1, \ldots, n_k \in \mathbb{N}$ such that $g = g_1^{n_1} \cdots g_k^{n_k}$ holds in $G$? Myasnikov, Nikolaev, and Ushakov proved that for every (Gromov-)hyperbolic group, the knapsack problem can be solved in polynomial time. In this paper, the precise complexity of the knapsack problem for hyperbolic group is dete…
We prove that the power word problem for certain metabelian subgroups of $\mathsf{GL}(2,\mathbb{C})$ (including the solvable Baumslag-Solitar groups $\mathsf{BS}(1,q) = \langle a,t \mid t a t^{-1} = a^q \rangle$) belongs to the circuit complexity class $\mathsf{TC}^0$. In the power word problem, the input consists of group elements $g_1, \ldots, g_d$ and binary encoded integers $n_1, \ldots, n_d$ and it is asked whether $g_1^{n_1} \cdots g_d^{n_d} = 1$ holds. Moreover, we prove that the knapsack problem for $\mathsf{BS}(1,q)$ is $\mathsf{NP}$-complete. In the knapsack problem, the input consis…
We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed $n \geq 3$, where $n$ is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks, which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and p…
We study relationship among versions of the Knapsack Problem where variables take values in Z and the number of them is fixed. In particular, we construct a finitely presented group where the problem of solvability of exponential equations with one variable is decidable but the corresponding problem for two variables is undecidable.
We address the problem of determining the class of self-similar groups, and in particular its closure under restricted direct products. We show that the group $\mathbb Z^{(ω)}$ is self-similar, that $G^{(ω)}\rtimes C_2$ is self-similar whenever $G$ is, and that permutational wreath products of a finite abelian group with a self-similar group are self-similar.
We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete automorphism group of the direct product.
We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C \wr F$, where $C$ is a finite group and $F$ a non-abelian free group.
Every semidirect product of groups $K\rtimes H$ has size $\left | K \right | \cdot \left | H\right |$, yet the size of such a group's automorphism group varies with the chosen action of $H$ on $K$. This paper will explore groups of the form $\text{Aut}(K\rtimes H)$, considering especially the case where $H$ and $K$ are cyclic. Only finite groups will be considered.
In a number of recent works, it has been established that many virtually free groups, almost all fundamental groups of surfaces and all groups which are nontrivial free products of groups satisfying a non-trivial law are algebraically closed in any group in which they are verbally closed. In this work we establish that any group which is a non-trivial free product is algebraically closed in any group in which it is verbally closed.
In this note, we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups.
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
For abelian groups $A, B$, $A$ is called $B$-small if the covariant functor $Hom(A,-)$ commutes with all direct sums $B^{(κ)}$ and $A$ is self-small provided it is $A$-small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.
We develop a theory of semidirect products of partial groups and localities. Our concepts generalize the notions of direct products of partial groups and localities, and of semidirect products of groups.
