2026-07-24T21:21:22Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:12533 2024-03-06T10:50:23Z ddc:004 open_access

The Complexity of Knapsack Problems in Wreath Products Figelius, Michael Ganardi, Moses Lohrey, Markus Zetzsche, Georg algorithmic group theory knapsack wreath product 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Michael Figelius and Moses Ganardi and Markus Lohrey and Georg Zetzsche 2020 Is Part Of LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2020.126 urn:nbn:de:0030-drops-125339 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.126 eng https://creativecommons.org/licenses/by/3.0/legalcode