Interdiction Games and Monotonicity, with Application to Knapsack Problems

AbstractTwo-person interdiction games represent an important modeling concept for applications in marketing, defending critical infrastructure, stopping nuclear weapons projects, or preventing drug smuggling. We present an exact branch-and-cut algorithm for interdiction games under the assumption that feasible solutions of the follower problem satisfy a certain monotonicity property. Prominent examples from the literature that fall into this category are knapsack interdiction, matching interdiction, and packing interdiction problems. We also show how practically relevant interdiction variants of facility location and prize-collecting problems can be modeled in our setting. Our branch-and-cut algorithm uses a solution scheme akin to Benders decomposition based on a family of so-called interdiction cuts. We present modified and lifted versions of these cuts along with exact and heuristic procedures for the separation of interdiction cuts and heuristic separation procedures for the other versions. In addition, we derive further valid inequalities and present a new heuristic procedure. We computationally evaluate the proposed algorithm on a benchmark of 360 knapsack interdiction instances from literature, including 27 instances for which the optimal solution was not known. Our approach is able to solve each of them to optimality within about one minute of computing time on a standard PC (in most cases, within just seconds), and it is up to some orders of magnitude faster than any previous approach from the literature. To further assess the effectiveness of our branch-and-cut algorithm, an additional computational study is performed on 144 randomly generated instances based on 0/1 multidimensional knapsack problems.