benchmark instances in [@CaCarLoWo16], requiring at most a computation time of about 85 seconds in an instance with 55 items. However, no computational evidence is provided in [@FiLjMoSi18] about the performance of the derived algorithm on larger instances. We also mention the work of [@FiMoSiEJOR18] where a heuristic approach is proposed for BKP and for other interdiction games. Other bilevel knapsack problems have been tackled in the literature. We mention the work in [@BrHaMa13] where the leader cannot interdict items but modifies the follower’s capacity. In [@ChZh13], the leader can modify the follower’s objective function only. As discussed in [@CaCarLoWo16], these knapsack problems are easier to handle than BKP. Recently, a polynomial algorithm has been provided in [@CaLoMa18] for the BKP variation where the follower solves a continuous knapsack problem.\
Our contribution for BKP is twofold. First, we derive effective lower bounds based on mathematical programming. Second, we present a new exact approach that exploits the induced follower’s problem and the derived lower bounds. The proposed approach shows up to be very effective successfully solving all benchmark literature instances provided in [@CaCarLoWo16] within few seconds of computation. Moreover, our algorithm manages to solve to optimality instances with up to 500 