Compressed Word Problems in HNN-extensions and Amalgamated Products

Haubold, Niko; Lohrey, Markus

doi:10.1007/s00224-010-9295-2

Cited by 16 publications

(11 citation statements)

References 18 publications

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“…This result was generalized to graph groups (right-angled Artin groups) [34] and fully residually free groups [36]. In [22], it was shown that the compressed word problem for an HNN-extension H, t | t −1 at = ϕ(a)(a ∈ A) with A a finite subgroup of the base group H can be reduced in polynomial time to the compressed word problem for H, and a corresponding result for amalgamated free products was shown as well.…”

Section: Introductionmentioning

confidence: 94%

Compressed Decision Problems for Graph Products and Applications to (Outer) Automorphism Groups

Haubold

Lohrey

Mathissen

2012

Int. J. Algebra Comput.

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It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a restricted variant of the simultaneous compressed conjugacy problem is polynomial time Turing-reducible to the same problem for the vertex groups. A direct corollary of this result is that the word problem for the outer automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Finally, it is shown that the compressed variant of the ordinary conjugacy problem can be solved in polynomial time for right-angled Artin groups.

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Section: Introductionmentioning

confidence: 94%

Compressed Decision Problems for Graph Products and Applications to (Outer) Automorphism Groups

Haubold

Lohrey

Mathissen

2012

Int. J. Algebra Comput.

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“…identified) subgroups, for which these constructions already play a prominent role, for example, in Stallings' decomposition of groups with infinitely many ends [41] or the construction of virtually free groups [9]. Moreover, these constructions are known to preserve a wide range of important structural and algorithmic properties [1,5,20,24,25,27,28,32,34,38]. ∈ Σ be a fresh generator.…”

Section: Part 2: Knapsack In Hnn-extensions and Amalgamated Productsmentioning

confidence: 99%

“…The following lemma provides a necessary and sufficient condition for equality of Britton-reduced words in an HNN-extension (cf. Lemma 2.2 of [20]):…”

Section: Part 2: Knapsack In Hnn-extensions and Amalgamated Productsmentioning

confidence: 99%

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Closure properties of knapsack semilinear groups

Figelius¹,

Lohrey²,

Zetzsche³

2019

Preprint

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We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form= g in a group G, where the variables x 1 , . . . , x k take values in the natural numbers): graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions. Moreover, we study the dependence of the so-called magnitude for the solution set of a knapsack equation (the magnitude is a complexity measure for semi-linear sets) with respect to the length of the knapsack equation (measured in number of generators). We investigate, how this dependence changes under the above group operations.This work has been supported by the DFG research project LO 748/13-1. 1 Karp in his seminal paper [26] defined knapsack in a slightly different way. NP-completeness of the above version was shown in [19].

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“…Compression techniques have been applied elsewhere for solving word problems, [14,15,25,11]. But in these papers the authors use straight-line programs whose compression rates are far too small (at best exponential) to cope with Baumslag or Higman groups.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Highly Compressed Data: The Word Problem in Higman's Group Is in P

Diekert

Laun

Ushakov

2012

Int. J. Algebra Comput.

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Abstract. Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the onerelator Baumslag group is in P. Before that the best known upper bound has been non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. The improvement is crucial for our other results. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.

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Compressed Word Problems in HNN-extensions and Amalgamated Products

Cited by 16 publications

References 18 publications

Compressed Decision Problems for Graph Products and Applications to (Outer) Automorphism Groups

Compressed Decision Problems for Graph Products and Applications to (Outer) Automorphism Groups

Closure properties of knapsack semilinear groups

Efficient Algorithms for Highly Compressed Data: The Word Problem in Higman's Group Is in P

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