follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s profits.\
In [@CaCarLoWo13] it is shown that BKP is $\Sigma_{2}^p$-complete in the polynomial hierarchy complexity. Essentially, BKP cannot be formulated as a single level problem unless the polynomial hierarchy collapses (as also pointed out in [@CaCarLoWo16]). This makes the problem even more difficult to solve than an NP-Complete problem. We refer to [@Je85] for an introduction on polynomial hierarchy.\
One of the best performing algorithms for BKP is given in [@CaCarLoWo16]. The algorithm, denoted as CCLW, relies on the dualization of the continuous relaxation of the follower’s problem and on iteratively computing upper bounds for the problem until a stopping criterion applies. The approach is motivated by the lack of significant lower bounds for the problem. Algorithm CCLW solves to optimality instances with 50 items within a CPU time limit of 3600 seconds, running out of time in instances with 55 items only. Very recently, an improved branch-and-cut algorithm has been given in [@FiLjMoSi18]. The proposed approach manages to solve to optimality all 