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Knapsack Problems for Wreath Products
[article]

2017
*
arXiv
*
pre-print

In recent years,

arXiv:1709.09598v2
fatcat:bv4mmiv32rgkpiprnlmn6jk5ai
*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. ...##
###
Knapsack Problems for Wreath Products

unpublished

In recent years,

fatcat:7v3tueoq4bgb7piefzqeywi6wu
*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*. ...##
###
The Complexity of Knapsack Problems in Wreath Products

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). ...

##
###
The complexity of knapsack problems in wreath products
[article]

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. ...

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

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

Moreover, we prove that the

doi:10.4230/lipics.mfcs.2020.67
dblp:conf/mfcs/LohreyZ20
fatcat:teyphoxj3bgi3ax6kq3eqp6524
*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). ...##
###
Knapsack and the power word problem in solvable Baumslag-Solitar groups
[article]

2020
*
arXiv
*
pre-print

Moreover, we prove that the

arXiv:2002.03837v2
fatcat:egqqcyqls5cuzgji7s2kgr5kda
*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). ...##
###
Contents
[chapter]

2020
*
Complexity and Randomness in Group Theory
*

inversion

doi:10.1515/9783110667028-toc
fatcat:hwlqmqii7zd6pfeew5ff6gm3ge
*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 ...##
###
Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

2019
*
Dagstuhl Reports
*

The Dagstuhl Seminar Algorithmic

doi:10.4230/dagrep.9.3.83
dblp:journals/dagstuhl-reports/DiekertKLM19
fatcat:7b3cwlpo6zdsxfzhiznob5cvsi
*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. ...##
###
Exponent Equations in HNN-extensions

2022
*
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
*

The class of

doi:10.1145/3476446.3535482
fatcat:3tapsx3hdbcyfa27ws3kxhgmdy
*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 ...##
###
Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups
[article]

2015
*
arXiv
*
pre-print

Moreover,

arXiv:1507.05145v1
fatcat:pb6nl6l2arezbiaezmn5ogqhre
*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. ...##
###
Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products

2016
*
Symposium on Theoretical Aspects of Computer Science
*

It is shown that the

doi:10.4230/lipics.stacs.2016.50
dblp:conf/stacs/LohreyZ16
fatcat:izbsudjb6fbkhg67szrcefx4ee
*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 ...##
###
Knapsack Problems in Groups
[article]

2013
*
arXiv
*
pre-print

We generalize the classical

arXiv:1302.5671v1
fatcat:krmzeuqdfzhtho3dewtggyph3q
*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. ...##
###
Knapsack problems in groups

2014
*
Mathematics of Computation
*

We generalize the classical

doi:10.1090/s0025-5718-2014-02880-9
fatcat:dq2m6tnocbb5zcuj72x7puvu3m
*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. ...##
###
Knapsack problems in products of groups

2016
*
Journal of symbolic computation
*

Our methods allow to obtain complexity results

doi:10.1016/j.jsc.2015.05.006
fatcat:7pbvhjqbozbgpnbpdq7sjc6gie
*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. ...##
###
Knapsack in Hyperbolic Groups
[chapter]

2018
*
Lecture Notes in Computer Science
*

over finite subgroups [17] , • direct

doi:10.1007/978-3-030-00250-3_7
fatcat:er47hhhqmjgwxh3ssp5hjeytkq
*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? ...
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