eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-29
126:1
126:18
10.4230/LIPIcs.ICALP.2020.126
article
The Complexity of Knapsack Problems in Wreath Products
Figelius, Michael
1
https://orcid.org/0000-0002-0407-0597
Ganardi, Moses
1
https://orcid.org/0000-0002-0775-7781
Lohrey, Markus
1
https://orcid.org/0000-0002-4680-7198
Zetzsche, Georg
2
https://orcid.org/0000-0002-6421-4388
Universität Siegen, Germany
Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable groups. For a finitely generated group we study the so-called power word problem (does a given expression u₁^{k₁} … u_d^{k_d}, where u₁, …, u_d are words over the group generators and k₁, …, k_d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u₁^{x₁} … u_d^{x_d} = v, where u₁, …, u_d,v are words over the group generators and x₁,…,x_d are variables, have a solution in the natural numbers). We prove that the power word problem for wreath products of the form G ≀ ℤ with G nilpotent and iterated wreath products of free abelian groups belongs to TC⁰. As an application of the latter, the power word problem for free solvable groups is in TC⁰. On the other hand we show that for wreath products G ≀ 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 coNP-hard. For the knapsack problem we show NP-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G ≀ ℤ, where G is uniformly efficiently non-solvable, is Σ₂^p-hard.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol168-icalp2020/LIPIcs.ICALP.2020.126/LIPIcs.ICALP.2020.126.pdf
algorithmic group theory
knapsack
wreath product