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Knapsack in Graph Groups

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Abstract

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.

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Notes

  1. Note that if we ask for a solution (x 1,…,x k ) in \(\mathbb {Z}^{k}\), then knapsack can be solved in polynomial time (even for binary encoded integers) by checking whether \(\gcd (g_{1}, \ldots , g_{k})\) divides g.

  2. In the following, T C 0 always refers to its DLOGTIME-uniform version.

  3. This term comes from the fact that right-angled Artin groups are exactly the Artin groups corresponding to right-angled Coxeter groups.

  4. 4Note that since alph(p i,j ) ⊆alph(u),we must have p i,j = 1or x k = 0whenever j < k < i.

  5. 5An arithmetic circuit is a finite directed acyclic graph, where every node of indegree zero is labeled with abinary encoded integer, and every node of non-zero indegree is labeled with one of the arithmetic operations+ or ×.Nodes (resp., edges) of the arithmetic circuit are also called gates (resp., wires) and there is adistinguished gate, called the output gate. Every gate evaluates to an integer (the value of thegate) in the natural way, and the arithmetic circuits evaluates to the value of its outputgate.

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Acknowledgments

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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Authors and Affiliations

  1. Universität Siegen, Siegen, Germany

    Markus Lohrey

  2. LSV, CNRS & ENS Paris-Saclay, Paris, France

    Georg Zetzsche

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  1. Markus Lohrey
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Correspondence to Markus Lohrey.

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This article is part of the Topical Collection on Theoretical Aspects of Computer Science

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Lohrey, M., Zetzsche, G. Knapsack in Graph Groups. Theory Comput Syst 62, 192–246 (2018). https://doi.org/10.1007/s00224-017-9808-3

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