Combinatorial Relaxations and Algorithms for Interdiction Problems

Abstract

In an interdiction problem, there are two players, the leader and the follower. The follower aims to solve some combinatorial optimization problem, such as knapsack, maximum matching, or shortest path. However, the leader, who acts before the follower, is able to pay some cost to destroy (i.e., interdict) parts of the structure that the follower's problem is solved over, subject to an interdiction budget. The objective is to determine the worst-case outcome for the follower. These problems are applied in a variety of scenarios including infrastructure reliability and planning, military defence, counter-trafficking, cybersecurity, medicine and public health.

The interdiction problems we consider are either $\Sigma_2^p$-complete or $\NP$-complete, so we cannot expect to find polynomial-time exact algorithms. Interdiction problems are considered a fundamental subclass of bilevel mixed-integer programming (BMIP). Although general-purpose BMIP solvers exist, they often struggle more on interdiction problems compared to other BMIP instances. A likely explanation is that the high-point relaxation, a standard BMIP relaxation which solvers rely on, is typically uninformative for interdiction problems. Hence, there is strong motivation for developing new relaxations which take advantage of the combinatorial structures present in interdiction problems.

This thesis develops new combinatorial relaxations and algorithms for interdiction problems, starting with knapsack interdiction. For this problem, we first develop a new relaxation using dynamic programming, which we utilize in a branch-and-bound scheme, achieving state-of-the-art computational results. Combining linear programming duality with dynamic programming, the following chapter gives a polynomial-time $(2+\epsilon)$-approximation for knapsack interdiction which is considerably simpler and faster than the known PTAS. Next, we consider matroid interdiction, where we introduce a new class of relaxations also built around dynamic programming. For partition matroids, this yields an exact pseudopolynomial time algorithm, and for graphic matroids, we use these bounds in a branch-and-bound scheme and achieve state-of-the-art computational results. We then unify these results in a general framework that derives bounds by reordering the players' decisions in a general two-player sequential game model. This recovers our earlier knapsack and matroid relaxations. Applied to matching interdiction, it yields new relaxations and a branch-and-bound algorithm which substantially outperforms branching on the natural linear programming relaxation.

The final chapter considers shortest path interdiction, where a fundamental obstacle prevents the practical application of our framework. Instead, we propose new combinatorial algorithms that enhance the Israeli-Wood Benders decomposition method. These enhancements improve computational performance by more than two orders of magnitude. The largest improvements are seen on large, dense graphs, a regime not addressed by prior works.

Together, these results establish a strong precedent for the use of combinatorial relaxations in solving interdiction problems. We hope that in future work these ideas can be applied more broadly and inform the next generation of general-purpose BMIP solvers.