Related papers: Knapsack Problems for Wreath Products
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…
We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable group. For a finitely generated group we study the so-called power word problem (does a given…
We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove…
It is shown that membership in rational subsets of wreath products H \wr V with H a finite group and V a virtually free group is decidable. On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath…
In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a…
We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time…
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…
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…
It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the…
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…
We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the power problem for $B$…
The knapsack problem is a classic optimisation problem that has been recently extended in the setting of groups. Its study reveals to be interesting since it provides many different behaviours, depending on the considered class of groups.…
We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form $g_1^{x_1} \cdots g_k^{x_k} = g$ in a group $G$, where the variables $x_1, \ldots, x_k$ take values…
We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to…
We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic…
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…
In this paper we study the complexity of solving orientable quadratic equations in wreath products $A\wr B$ of finitely generated abelian groups. We give a classification of cases (depending on genus and other characteristics of a given…
We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been…
It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form…
Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In previous work, the authors proved that for each graph group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine the exact…