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Knapsack Problems for Wreath Products [article]

Moses Ganardi, Daniel König, Markus Lohrey, Georg Zetzsche
2017 arXiv   pre-print
In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied.  ...  Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem) for the wreath product G Z is NP-complete.  ...  In this paper, we study the knapsack problem for wreath products. The wreath product is a fundamental construction in group theory and semigroup theory, see Section 4 for the definition.  ... 
arXiv:1709.09598v2 fatcat:bv4mmiv32rgkpiprnlmn6jk5ai

Knapsack Problems for Wreath Products

Moses Ganardi, Daniel König, Markus Lohrey, Georg Zetzsche
unpublished
In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied.  ...  Finally, it is shown that for every non-trivial abelian group G, knapsack (as well as the related subset sum problem) for the wreath product G Z is NP-complete.  ...  We begin with a general necessary condition for knapsack to be decidable for a wreath product.  ... 
fatcat:7v3tueoq4bgb7piefzqeywi6wu

The Complexity of Knapsack Problems in Wreath Products

Michael Figelius, Moses Ganardi, Markus Lohrey, Georg Zetzsche, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
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.  ...  126:16 The Complexity of Knapsack Problems in Wreath Products of the variables to elements of G (called EqnId(G) in [44] ) can be easily reduced to ∀-Sat(G).  ... 
doi:10.4230/lipics.icalp.2020.126 dblp:conf/icalp/FigeliusGLZ20 fatcat:ptqydcxlknaavoiyhgm42dofci

The complexity of knapsack problems in wreath products [article]

Michael Figelius, Moses Ganardi, Markus Lohrey, Georg Zetzsche
2020 arXiv   pre-print
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 Σ^2_p-hard.  ...  C V I T 2 0 1 6 23:18 The complexity of knapsack problems in wreath products Proof.  ... 
arXiv:2002.08086v1 fatcat:xcrnxqt4irftxjvfdl7nt5r7mi

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

Markus Lohrey, Georg Zetzsche, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete.  ...  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.  ...  Our algorithm for the knapsack problem in BS(1, q) cannot be extended to solvability of exponent equations (not even to solvability of a single exponent equation).  ... 
doi:10.4230/lipics.mfcs.2020.67 dblp:conf/mfcs/LohreyZ20 fatcat:teyphoxj3bgi3ax6kq3eqp6524

Knapsack and the power word problem in solvable Baumslag-Solitar groups [article]

Markus Lohrey, Georg Zetzsche
2020 arXiv   pre-print
Moreover, we prove that the knapsack problem for BS(1,q) is NP-complete.  ...  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 TC^0.  ...  Our algorithm for the knapsack problem in BS(1, q) cannot be extended to solvability of exponent equations (not even to solvability of a single exponent equation).  ... 
arXiv:2002.03837v2 fatcat:egqqcyqls5cuzgji7s2kgr5kda

Contents [chapter]

2020 Complexity and Randomness in Group Theory  
inversion problem | 335 6.9 Semidirect product of groups and more peculiar computational assumptions | 339 6.10 The subset sum problem and the knapsack problem | 342 6.11 The hidden subgroup problem |  ...  problem for Aut(F 3 ) | 169 4.6.2 Straight-line programs | 170 4.6.3 The Diffie-Hellman key exchange protocol | 318 6.2.1 1 Motivation: the word problem for the Baumslag group | 199 4.7.2 Power  ... 
doi:10.1515/9783110667028-toc fatcat:hwlqmqii7zd6pfeew5ff6gm3ge

Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

Volker Diekert, Olga Kharlampovich, Markus Lohrey, Alexei Myasnikov, Michael Wagner
2019 Dagstuhl Reports  
The Dagstuhl Seminar Algorithmic Problems in Group Theory was aimed at bringing together researchers from group theory and computer science so that they can share their expertise.  ...  Moses Ganardi talked on wreath products as well, but put the focus on the knapsack problem.  ...  ., 3 Knapsack problems for wreath products Moses Ganardi (Universität Siegen, DE) Word equations are an important problem on the intersection of formal languages and algebra.  ... 
doi:10.4230/dagrep.9.3.83 dblp:journals/dagstuhl-reports/DiekertKLM19 fatcat:7b3cwlpo6zdsxfzhiznob5cvsi

Exponent Equations in HNN-extensions

Michael Figelius, Markus Lohrey
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
The class of knapsack semilinear groups is quite rich and it is closed under many group theoretic constructions, e.g., finite extensions, graph products, wreath products, amalgamated free products with  ...  Solvability of such (systems of) equations has been intensively studied for various classes of groups in recent years.  ...  is closed under the following operations: finite extensions [10] , graph products [10] , wreath products [12] , amalgamated free products with finite amalgamated subgroups [10] , and HNN-extensions  ... 
doi:10.1145/3476446.3535482 fatcat:3tapsx3hdbcyfa27ws3kxhgmdy

Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups [article]

Daniel König, Markus Lohrey, Georg Zetzsche
2015 arXiv   pre-print
Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products.  ...  It is also shown that for every co-context-free group, the knapsack problem is decidable.  ...  Knapsack problems for finite extensions We show that in contrast to direct products, decidability of the knapsack problem is preserved under finite extensions.  ... 
arXiv:1507.05145v1 fatcat:pb6nl6l2arezbiaezmn5ogqhre

Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products

Markus Lohrey, Georg Zetzsche, Marc Herbstritt
2016 Symposium on Theoretical Aspects of Computer Science  
It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups.  ...  We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified  ...  For the following groups, subset sum is NP-complete (whereas the word problem can be solved in polynomial time): free metabelian non-abelian groups of finite rank, the wreath product Z Z, Thompson's group  ... 
doi:10.4230/lipics.stacs.2016.50 dblp:conf/stacs/LohreyZ16 fatcat:izbsudjb6fbkhg67szrcefx4ee

Knapsack Problems in Groups [article]

Alexei Myasnikov, Andrey Nikolaev, Alexander Ushakov
2013 arXiv   pre-print
We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems.  ...  We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem  ...  (b) Wreath product Z ≀ Z. (c) Wreath product of two finitely generated infinite abelian groups. Proposition 4 . 4 . 44 The subset sum problem for the Thompson's group F is NPcomplete.  ... 
arXiv:1302.5671v1 fatcat:krmzeuqdfzhtho3dewtggyph3q

Knapsack problems in groups

Alexei Myasnikov, Andrey Nikolaev, Alexander Ushakov
2014 Mathematics of Computation  
We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems.  ...  We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem  ...  (b) Wreath product Z ≀ Z. (c) Wreath product of two finitely generated infinite abelian groups. Proposition 4 . 4 . 44 The subset sum problem for the Thompson's group F is NPcomplete.  ... 
doi:10.1090/s0025-5718-2014-02880-9 fatcat:dq2m6tnocbb5zcuj72x7puvu3m

Knapsack problems in products of groups

Elizaveta Frenkel, Andrey Nikolaev, Alexander Ushakov
2016 Journal of symbolic computation  
Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.  ...  We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it.  ...  Myasnikov for bringing their attention to the problem and for insightful advice and discussions, and to the anonymous referees for their astute observations and helpful suggestions.  ... 
doi:10.1016/j.jsc.2015.05.006 fatcat:7pbvhjqbozbgpnbpdq7sjc6gie

Knapsack in Hyperbolic Groups [chapter]

Markus Lohrey
2018 Lecture Notes in Computer Science  
over finite subgroups [17] , • direct products (in contrast, the class of groups with a decidable knapsack problem is not closed under direct products), • restricted wreath products [5] .  ...  The knapsack problem for G is the following decision problem: Input: A single knapsack expression E over G. Question: Is sol(E) non-empty?  ... 
doi:10.1007/978-3-030-00250-3_7 fatcat:er47hhhqmjgwxh3ssp5hjeytkq
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