Knapsack Problems for Wreath Products

TL;DR

This paper investigates the decidability and complexity of the knapsack problem in wreath products of groups, revealing that decidability is not preserved but certain classes remain decidable, with some cases being NP-complete.

Contribution

It demonstrates that decidability of knapsack problems is not preserved under wreath products and identifies classes where it remains decidable, such as free solvable groups.

Findings

Decidability of knapsack is not preserved under wreath product.

Knapsack-semilinear groups are closed under wreath product.

Knapsack for G wr Z is NP-complete for any non-trivial abelian G.

Abstract

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 $G \wr \mathbb{Z}$ is NP-complete.