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.ICALP.2023.134
URN: urn:nbn:de:0030-drops-181864
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18186/
Go to the corresponding LIPIcs Volume Portal

Lohrey, Markus ; Rosowski, Andreas

On the Complexity of Diameter and Related Problems in Permutation Groups

pdf-format:
LIPIcs-ICALP-2023-134.pdf (0.8 MB)

Abstract

We prove that it is Π₂^?-complete to verify whether the diameter of a given permutation group G = ⟨A⟩ is bounded by a unary encoded number k. This solves an open problem from a paper of Even and Goldreich, where the problem was shown to be NP-hard. Verifying whether the diameter is exactly k is complete for the class consisting of all intersections of a Π₂^?-language and a Σ₂^?-language. A similar result is shown for the length of a given permutation π, which is the minimal k such that π can be written as a product of at most k generators from A. Even and Goldreich proved that it is NP-complete to verify, whether the length of a given π is at most k (with k given in unary encoding). We show that it is DP-complete to verify whether the length is exactly k. Finally, we deduce from our result on the diameter that it is Π₂^?-complete to check whether a given finite automaton with transitions labelled by permutations from S_n produces all permutations from S_n.

BibTeX - Entry

@InProceedings{lohrey_et_al:LIPIcs.ICALP.2023.134,
  author =	{Lohrey, Markus and Rosowski, Andreas},
  title =	{{On the Complexity of Diameter and Related Problems in Permutation Groups}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{134:1--134:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18186},
  URN =		{urn:nbn:de:0030-drops-181864},
  doi =		{10.4230/LIPIcs.ICALP.2023.134},
  annote =	{Keywords: algorithms for finite groups, diameter of permutation groups, rational subsets in groups}
}

Keywords: algorithms for finite groups, diameter of permutation groups, rational subsets in groups
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI