Lower Bounds and a New Exact Approach for the Bilevel Knapsack with Interdiction Constraints

Abstract

We consider the Bilevel Knapsack with Interdiction Constraints, an extension of the classic 0-1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the 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. The presence of two decision levels makes this problem very difficult to solve in practice: the current state-of-the-art algorithms can solve to optimality instances with 50–55 items at most. We derive effective lower bounds and present a new exact approach that exploits the structure of the induced follower’s problem. The approach successfully solves all benchmark instances within one second in the worst case and larger instances with up to 500 items within 60 s.