eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-08-22
71:1
71:16
10.4230/LIPIcs.MFCS.2022.71
article
Membership Problems in Finite Groups
Lohrey, Markus
1
https://orcid.org/0000-0002-4680-7198
Rosowski, Andreas
1
Zetzsche, Georg
2
https://orcid.org/0000-0002-6421-4388
Universität Siegen, Germany
Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
We show that the subset sum problem, the knapsack problem and the rational subset membership problem for permutation groups are NP-complete. Concerning the knapsack problem we obtain NP-completeness for every fixed n ≥ 3, where n is the number of permutations in the knapsack equation. In other words: membership in products of three cyclic permutation groups is NP-complete. This sharpens a result of Luks [Eugene M. Luks, 1991], which states NP-completeness of the membership problem for products of three abelian permutation groups. We also consider the context-free membership problem in permutation groups and prove that it is PSPACE-complete but NP-complete for a restricted class of context-free grammars where acyclic derivation trees must have constant Horton-Strahler number. Our upper bounds hold for black box groups. The results for context-free membership problems in permutation groups yield new complexity bounds for various intersection non-emptiness problems for DFAs and a single context-free grammar.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol241-mfcs2022/LIPIcs.MFCS.2022.71/LIPIcs.MFCS.2022.71.pdf
algorithms for finite groups
intersection non-emptiness problems
knapsack problems in groups