Knapsack Problems for Wreath Products

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem) for the wreath product G wr Z is NP-complete

The Complexity of Knapsack Problems in Wreath Products

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 expression $u_1^{k_1} \ldots u_d^{k_d}$, where $u_1, \ldots, u_d$ are words over the group generators and $k_1, \ldots, k_d$ are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation $u_1^{x_1} \ldots u_d^{x_d} = v$, where $u_1, \ldots, u_d,v$ are words over the group generators and $x_1,\ldots,x_d$ are variables, has a solution in the natural numbers). We prove that the power word problem for wreath products of the form $G \wr \mathbb{Z}$ with $G$ nilpotent and iterated wreath products of free abelian groups belongs to $\mathsf{TC}^0$. As an application of the latter, the power word problem for free solvable groups is in $\mathsf{TC}^0$. On the other hand we show that for wreath products $G \wr \mathbb{Z}$, where $G$ is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is $\mathsf{coNP}$-hard. For the knapsack problem we show $\mathsf{NP}$-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product $G \wr \mathbb{Z}$, where $G$ is uniformly efficiently non-solvable, is $\Sigma^2_p$-hard

A Characterization of Wreath Products Where Knapsack Is Decidable

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 introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to $H_3(\mathbb{Z})$, the discrete Heisenberg group, and to Baumslag-Solitar groups $\mathsf{BS}(1,q)$ for $q\ge 1$. First, we show that the knapsack problem is undecidable for $G\wr H_3(\mathbb{Z})$ for any $G\ne 1$. This implies that for $G\ne 1$ and for infinite and virtually nilpotent groups $H$, the knapsack problem for $G\wr H$ is decidable if and only if $H$ is virtually abelian and solvability of systems of exponent equations is decidable for $G$. Second, we show that the knapsack problem is decidable for $G\wr\mathsf{BS}(1,q)$ if and only if solvability of systems of exponent equations is decidable for $G$

Knapsack Problems in Groups

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 decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure

Knapsack problems in products of groups

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.Comment: 15 pages, 5 figures. Updated to include more general results, mostly in Section