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Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups

Authors Markus Lohrey , Georg Zetzsche

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Markus Lohrey
  • Universität Siegen, Germany
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

Funding

  • Lohrey, Markus: Funded by DFG project LO 748/12-1.

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Markus Lohrey and Georg Zetzsche. Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.67

Abstract

We prove that the power word problem for the solvable Baumslag-Solitar groups BS(1,q) = ⟨ a,t ∣ t a t^{-1} = a^q ⟩ can be solved in TC⁰. In the power word problem, the input consists of group elements g₁, …, g_d and binary encoded integers n₁, …, n_d and it is asked whether g₁^{n₁} ⋯ g_d^{n_d} = 1 holds. Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete. In the knapsack problem, the input consists of group elements g₁, …, g_d,h and it is asked whether the equation g₁^{x₁} ⋯ g_d^{x_d} = h has a solution in ℕ^d.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • computational group theory
  • matrix problems
  • Baumslag-Solitar groups

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