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Knapsack Problems for Wreath Products
[article]
2017
arXiv
pre-print
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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
unpublished
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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
2020
International Colloquium on Automata, Languages and Programming
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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]
2020
arXiv
pre-print
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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
2020
International Symposium on Mathematical Foundations of Computer Science
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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]
2020
arXiv
pre-print
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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
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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)
2019
Dagstuhl Reports
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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
2022
Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation
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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]
2015
arXiv
pre-print
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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
2016
Symposium on Theoretical Aspects of Computer Science
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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]
2013
arXiv
pre-print
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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
2014
Mathematics of Computation
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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
2016
Journal of symbolic computation
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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]
2018
Lecture Notes in Computer Science
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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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