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Knapsack in Hyperbolic Groups

  • Markus LohreyEmail author
Conference paper
  • 141 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11123)

Abstract

Recently knapsack problems have been generalized from the integers to arbitrary finitely generated groups. The knapsack problem for a finitely generated group G is the following decision problem: given a tuple \((g, g_1, \ldots , g_k)\) of elements of G, are there natural numbers \(n_1, \ldots , n_k \in \mathbb {N}\) such that \(g = g_1^{n_1} \cdots g_k^{n_k}\) holds in G? Myasnikov, Nikolaev, and Ushakov proved that for every hyperbolic group, the knapsack problem can be solved in polynomial time. In this paper, it is shown that for every hyperbolic group G, the knapsack problem belongs to the complexity class \(\mathsf {LogCFL}\), and it is \(\mathsf {LogCFL}\)-complete if G contains a free group of rank two. Moreover, it is shown that for every hyperbolic group G and every tuple \((g, g_1, \ldots , g_k)\) of elements of G the set of all \((n_1, \ldots , n_k) \in \mathbb {N}^k\) such that \(g = g_1^{n_1} \cdots g_k^{n_k}\) in G is effectively semilinear.

Keywords

Hyperbolic Groups Knapsack Problem Myasnikov Semilinear Representation Semilinear Set 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

This work has been supported by the DFG research project LO 748/13-1.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universität SiegenSiegenGermany

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