  The knapsack problem is a classic optimisation problem that has been recently
extended in the setting of groups. Its study reveals to be interesting since it
provides many different behaviours, depending on the considered class of
groups. In this paper we deal with groups generated by Mealy automata-a class
that is often used to study group-theoretical conjectures-and prove that the
knapsack problem is undecidable for this class. In a second time, we construct
a graph that, if finite, provides a solution to the knapsack problem. We deduce
that the knapsack problem is decidable for the so-called bounded automaton
groups, a class where the order and conjugacy problems are already known to be
decidable.
