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License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2021.11
URN: urn:nbn:de:0030-drops-136566
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13656/
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Bergsträßer, Pascal ; Ganardi, Moses ; Zetzsche, Georg

A Characterization of Wreath Products Where Knapsack Is Decidable

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LIPIcs-STACS-2021-11.pdf (0.8 MB)

Abstract

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,…,g_n,g ∈ G and asks whether there are x_1,…,x_n ≥ 0 with g_1^{x_1}⋯ g_n^{x_n} = g. We study the knapsack problem for wreath products G≀H of groups G and H.
Our main result is a characterization of those wreath products G≀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 H₃(ℤ), the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1,q) for q ≥ 1. First, we show that the knapsack problem is undecidable for G≀H₃(ℤ) for any G ≠ 1. This implies that for G ≠ 1 and for infinite and virtually nilpotent groups H, the knapsack problem for G≀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≀BS(1,q) if and only if solvability of systems of exponent equations is decidable for G.

BibTeX - Entry

@InProceedings{bergstraer_et_al:LIPIcs.STACS.2021.11,
  author =	{Bergstr\"{a}{\ss}er, Pascal and Ganardi, Moses and Zetzsche, Georg},
  title =	{{A Characterization of Wreath Products Where Knapsack Is Decidable}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13656},
  URN =		{urn:nbn:de:0030-drops-136566},
  doi =		{10.4230/LIPIcs.STACS.2021.11},
  annote =	{Keywords: knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups}
}

Keywords: knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups
Collection: 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)
Issue Date: 2021
Date of publication: 10.03.2021

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