The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

Miasnikov, Alexei; Vassileva, Svetla; Weiß, Armin

doi:10.1007/s00224-018-9849-2

Cited by 7 publications

(11 citation statements)

References 29 publications

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“…Note that the above NP-hardness proof also works for the subset sum problem, where the range of the valuation is restricted to {0, 1}. Moreover, if the word problems for two groups G and H can be solved in polynomial time, then word problem for G ≀ H can be solved in polynomial time as well [21]. This implies that subset sum for G ≀ H belongs to NP.…”

Section: Case 1 Consider An Expression U XImentioning

confidence: 96%

“…Complete problems for TC 0 are multiplication and division of binary encoded integers (or, more precisely, the question whether a certain bit in the output number is 1) [10]. TC 0 -complete problems in the context of group theory are the word problem for any infinite finitely generated solvable linear group [13], the subgroup membership problem for finitely generated nilpotent groups [25], the conjugacy problem for free solvable groups and wreath products of abelian groups [21], and the knapsack problem for finitely generated abelian groups [19].…”

Section: Complexity: Proof Of Theorem 57mentioning

confidence: 99%

“…An important application of wreath products in group theory is the Magnus embedding theorem [20], which allows to embed the quotient group F k /[N, N ] into the wreath product Z k ≀ (F k /N ), where F k is a free group of rank k and N is a normal subgroup of F k . From the algorithmic point of view, wreath products have some nice properties: The word problem for a wreath product G ≀ H is AC 0 -reducible to the word problems for the factors G and H, and the conjugacy problem for G ≀ H is TC 0 -reducible to the conjugacy problems for G and H and the so called power problem for H [21].…”

Section: Introductionmentioning

confidence: 99%

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Knapsack Problems for Wreath Products

Ganardi,

König,

Lohrey

et al. 2017

Preprint

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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. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under wreath product. As a consequence, we obtain the decidability of knapsack for free solvable groups. 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.

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Section: Case 1 Consider An Expression U XImentioning

confidence: 96%

Section: Complexity: Proof Of Theorem 57mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Knapsack Problems for Wreath Products

Ganardi,

König,

Lohrey

et al. 2017

Preprint

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“…It was only recently shown in [35] that the word problem (as well as the conjugacy problem) for every free solvable group belongs to TC 0 . Theorem 3 generalizes this result (at least the part on the word problem).…”

Section: :4mentioning

confidence: 99%

The complexity of knapsack problems in wreath products

Figelius¹,

Ganardi²,

Lohrey³

et al. 2020

Preprint

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We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable group. For a finitely generated group we study the so-called power word problem (does a given expression u k 1 1 . . . u k d d , where u1, . . . , u d are words over the group generators and k1, . . . , k d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u x 1 1 . . . u x d d = v, where u1, . . . , u d , v are words over the group generators and x1, . . . , x d are variables, has a solution in the natural numbers). We prove that the power word problem for wreath products of the form G Z with G nilpotent and iterated wreath products of free abelian groups belongs to TC 0 . As an application of the latter, the power word problem for free solvable groups is in TC 0 . On the other hand we show that for wreath products G Z, where G is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is coNP-hard. 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.2012 ACM Subject Classification CCS → Theory of computation → computational complexity and cryptography → problems, reductions and completeness Keywords and phrases algorithmic group theory, knapsack, wreath product

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“…This has been used by several authors to obtain algorithms for groups of the form F/[N, N ], and in particular free solvable groups. Examples include the word problem (folklore, see [15]), the conjugacy problem [15,28,29,30], the power problem [15], and the knapsack problem [11,12].…”

Section: Introductionmentioning

confidence: 99%

A characterization of wreath products where knapsack is decidable

Bergsträßer¹,

Ganardi²,

Zetzsche³

2021

Preprint

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The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group G and takes as input group elements g1, . . . , gn, g ∈ G and asks whether there are x1, . . . , xn ≥ 0 withWe study the knapsack problem for wreath products G ≀ H of groups G and H.Our main result is a characterization of those wreath products G ≀ H for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors G and H. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem.Moreover, we apply our main result to H3(Z), the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1, q) for q ≥ 1. First, we show that the knapsack problem is undecidable for G ≀ H3(Z) for any G ̸ = 1. This implies that for G ̸ = 1 and for infinite and virtually nilpotent groups H, the knapsack problem for G ≀ H is decidable if and only if H is virtually abelian and solvability of systems of exponent equations is decidable for G. Second, we show that the knapsack problem is decidable for G ≀ BS(1, q) if and only if solvability of systems of exponent equations is decidable for G. ACM Subject ClassificationTheory of computation → Problems, reductions and completeness; Theory of computation → Theory and algorithms for application domains Keywords and phrases knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups Digital Object Identifier 10.4230/LIPIcs...

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The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

Cited by 7 publications

References 29 publications

Knapsack Problems for Wreath Products

Knapsack Problems for Wreath Products

The complexity of knapsack problems in wreath products

A characterization of wreath products where knapsack is decidable

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