The Complexity of Knapsack in Graph Groups

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In LohreyZ16 the authors proved that for each graph group, the knapsack problem can be solved in NP. Here, we determine the exact complexity of the problem for every graph group. While the problem is TC^0-complete for complete graphs, it is LogCFL-complete for each (non-complete) transitive forest. For every remaining graph, the problem is NP-complete

Knapsack Problems for Wreath Products

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 wr Z is NP-complete