A fast combinatorial algorithm for the bilevel knapsack problem with interdiction constraints

We consider the bilevel knapsack problem with interdiction constraints, a fundamental bilevel integer programming problem which generalizes the 0–1 knapsack problem. In this problem, there are two knapsacks and n items. The objective is to select some items to pack into the first knapsack such that the maximum profit attainable from packing some of the remaining items into the second knapsack is minimized. We present a combinatorial branch-and-bound algorithm which outperforms the current state-of-the-art solution method in computational experiments for 99% of the instances reported in the literature. On many of the harder instances, our algorithm is orders of magnitude faster, which enabled it to solve 53 of the 72 previously unsolved instances. Our result relies fundamentally on a new dynamic programming algorithm which computes very strong lower bounds. This dynamic program solves a relaxation of the problem from bilevel to 2n-level where the items are processed in an online fashion. The relaxation is easier to solve but approximates the original problem surprisingly well in practice. We believe that this same technique may be useful for other interdiction problems.