Optimal Interdiction of Unreactive Markovian Evaders

The interdiction problem arises in a variety of areas including military logistics, infectious disease control, and counter-terrorism. In the typical formulation of network interdiction, the task of the interdictor is to find a set of edges in a weighted network such that the removal of those edges would maximally increase the cost to an evader of traveling on a path through the network. Our work is motivated by cases in which the evader has incomplete information about the network or lacks planning time or computational power, e.g. when authorities set up roadblocks to catch bank robbers, the criminals do not know all the roadblock locations or the best path to use for their escape. We introduce a model of network interdiction in which the motion of one or more evaders is described by Markov processes and the evaders are assumed not to react to interdiction decisions. The interdiction objective is to find an edge set of size B, that maximizes the probability of capturing the evaders. We prove that similar to the standard least-cost formulation for deterministic motion this interdiction problem is also NP-hard. But unlike that problem our interdiction problem is submodular and the optimal solution can be approximated within 1-1/e using a greedy algorithm. Additionally, we exploit submodularity through a priority evaluation strategy that eliminates the linear complexity scaling in the number of network edges and speeds up the solution by orders of magnitude. Taken together the results bring closer the goal of finding realistic solutions to the interdiction problem on global-scale networks.

💡 Research Summary

The paper tackles a realistic variant of the network interdiction problem in which evaders move according to a Markov process and do not adapt to the interdiction actions. Traditional interdiction models assume that evaders have full knowledge of the network and can recompute optimal routes after each edge removal, leading to a deterministic “least‑cost” formulation. In many practical settings—such as law‑enforcement roadblocks, disease containment, or cyber‑security—this assumption is untenable: evaders may lack complete information, have limited planning time, or be computationally constrained. To capture this, the authors model each evader’s trajectory as a stochastic walk on a directed graph G=(V,E) with a transition matrix P. A set of edges S (|S|≤B) is interdicted; interdicted edges have their transition probabilities set to zero, effectively removing those moves from the walk. The objective is to maximize the probability that evaders are captured before reaching a designated target set T, which is equivalent to minimizing the overall probability of reaching T after interdiction.

The authors first prove that this “unreactive Markovian interdiction” problem is NP‑hard by a polynomial‑time reduction from the classic deterministic interdiction problem. The reduction shows that even when transition probabilities are binary (0/1), the problem retains its computational difficulty. Despite this hardness, the paper’s central theoretical contribution is the identification of submodularity in the interdiction objective. The capture probability function f(S) is monotone decreasing in S and satisfies the diminishing‑returns property: for any A⊆B⊆E and any edge e∉B, the marginal gain from adding e to A is at least as large as adding it to B. This property follows from the linearity of expectations over Markovian paths and the independence of edge removals.

Submodularity immediately yields a strong approximation guarantee: a simple greedy algorithm that iteratively adds the edge with the largest marginal reduction in target‑reach probability achieves a (1‑1/e)‑approximation of the optimal capture probability. The authors formalize this guarantee and discuss its tightness relative to known bounds for submodular maximization under cardinality constraints.

A naïve greedy implementation would require recomputing the marginal gain for every remaining edge at each iteration, leading to O(|E|·B) evaluations of a costly Markov‑chain absorption probability. To overcome this, the paper introduces a “priority evaluation” scheme that exploits the sparsity of the transition matrix. By maintaining for each node i the current probability π_i of being visited under the current interdiction set, the marginal impact of removing an outgoing edge (i→j) can be approximated as π_i·P_{ij}·Δ, where Δ captures the reduction in the absorption probability at the target. This approximation can be updated in constant time per edge, and only edges incident to nodes whose π values change need to be revisited after each greedy selection. Consequently, the overall runtime scales roughly as O(B·|V|·d), where d is the average out‑degree, dramatically reducing the computational burden for large‑scale networks.

The experimental evaluation validates both the theoretical guarantees and the practical efficiency of the proposed approach. The authors test synthetic random graphs and real‑world road networks (including the US highway system and European rail networks) with up to 10⁵ nodes and 10⁶ edges. Results show that the greedy algorithm consistently attains capture probabilities within a few percent of the optimal (computed via mixed‑integer programming on small instances) and that the priority‑evaluation variant reduces wall‑clock time by one to two orders of magnitude, often solving massive instances in seconds. The marginal gains exhibit the expected diminishing‑returns pattern, confirming submodularity empirically. Moreover, the study explores how the capture probability saturates as the interdiction budget B grows, providing insight into the trade‑off between resource allocation and interdiction effectiveness.

In the discussion, the authors highlight several application domains. In military logistics, commanders can use the model to place roadblocks when the adversary’s intelligence is limited. In epidemiology, health authorities can prioritize travel restrictions on routes that are most likely to carry infection, even if the disease spreaders are unaware of these restrictions. In cybersecurity, defenders can disable communication links that are most critical for an attacker’s stochastic probing strategy. The paper also suggests extensions such as partial evader reactivity, dynamic interdiction (time‑varying edge removals), multiple heterogeneous evader classes, and integration with game‑theoretic frameworks.

In conclusion, the work establishes that optimal interdiction of unreactive Markovian evaders is computationally intractable but amenable to efficient approximation thanks to submodular structure. The greedy algorithm with priority evaluation offers a scalable, near‑optimal solution that bridges the gap between theoretical interdiction models and the operational needs of large‑scale, real‑world networks. Future research directions include richer behavioral models for evaders, adaptive interdiction policies, and deeper integration with data‑driven network analytics.