Theory of Computing Systems

, Volume 62, Issue 1, pp 192–246 | Cite as

Knapsack in Graph Groups

  • Markus Lohrey
  • Georg Zetzsche
Part of the following topical collections:
  1. Theoretical Aspects of Computer Science


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 every graph group. This result even holds if the group elements are represented in a compressed form by so called straight-line programs, which generalizes the classical NP-completeness result of the integer knapsack problem. If group elements are represented explicitly by words over the generators, then knapsack for a graph group belongs to the class LogCFL (a subclass of P) if the graph group can be built up from the trivial group using the operations of free product and direct product with \(\mathbb {Z}\). In all other cases, the knapsack problem is NP-complete.


Graph groups Knapsack problems Combinatorial group theory Decision problems in group theory 



Georg Zetzsche is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD) and by Labex Digicosme, Univ. Paris-Saclay, project VERICONISS. Markus Lohrey is supported by the DFG project LO 748/12-1.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Universität SiegenSiegenGermany
  2. 2.LSV, CNRS & ENS Paris-SaclayParisFrance

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