Interdiction of a Markovian Evader

Shortest path network interdiction is a combinatorial optimization problem on an activity network arising in a number of important security-related applications. It is classically formulated as a bilevel maximin problem representing an “interdictor” and an “evader”. The evader tries to move from a source node to the target node along a path of the least cost while the interdictor attempts to frustrate this motion by cutting edges or nodes. The interdiction objective is to find the optimal set of edges to cut given that there is a finite interdiction budget and the interdictor must move first. We reformulate the interdiction problem for stochastic evaders by introducing a model in which the evader follows a Markovian random walk guided by the least-cost path to the target. This model can represent incomplete knowledge about the evader, and the resulting model is a nonlinear 0-1 optimization problem. We then introduce an optimization heuristic based on betweenness centrality that can rapidly find high-quality interdiction solutions by providing a global view of the network.

💡 Research Summary

The paper revisits the classic shortest‑path network interdiction problem, which is traditionally cast as a bilevel max‑min game between an interdictor and an evader. In the conventional formulation the evader is assumed to have perfect knowledge of the network and to travel deterministically along the cheapest s‑t path after the interdictor has removed a subset of arcs subject to a budget constraint. This deterministic assumption, however, is rarely realistic in security‑related contexts where the evader’s information may be incomplete, its behavior may be stochastic, or it may deliberately randomize its route to avoid detection.

To capture this uncertainty the authors introduce a stochastic evader model based on a Markovian random walk that is “guided” by the underlying cost structure. For each node i, the transition probability to a neighbor j is defined as

 p_{ij}=exp(−α·c_{ij}) / Σ_{k∈N(i)} exp(−α·c_{ik}),

where c_{ij} is the traversal cost of arc (i,j) and α≥0 controls the sensitivity to cost. When an arc is interdicted its cost is set to infinity, which forces the corresponding transition probability to zero. Consequently, the evader’s trajectory is a probability distribution over all feasible s‑t walks, with a bias toward low‑cost routes but retaining a non‑zero chance of taking sub‑optimal edges.

Under this stochastic motion the evader’s objective becomes the minimization of expected travel cost (or equivalently, the maximization of the probability of reaching the target within a given cost budget). The interdictor’s problem is to choose a binary vector x∈{0,1}^{|E|} indicating which arcs to cut, subject to a linear budget Σ_{e} w_e x_e ≤ B (w_e is the interdiction cost of arc e). The objective is to maximize the evader’s expected cost after interdiction. Because the transition probabilities p_{ij}(x) depend non‑linearly on x, the resulting formulation is a non‑linear 0‑1 integer program, which is NP‑hard and intractable for large networks.

Recognizing the computational difficulty, the authors propose a heuristic that exploits a global structural measure: edge betweenness centrality. Betweenness quantifies how many shortest‑path routes between all node pairs pass through a given edge. The intuition is that edges with high betweenness are critical conduits for low‑cost travel, and removing them is likely to disrupt many potential evader routes. The heuristic proceeds as follows:

1. Compute betweenness scores for all edges in the original network.
2. For each candidate edge e, estimate the marginal increase ΔC(e) in the evader’s expected cost if e were interdicted, given the current set of interdicted edges X. This can be done by recomputing the Markov transition matrix after temporarily removing e and evaluating the new expected cost.
3. Rank edges by the ratio ΔC(e)/w_e (expected‑cost gain per unit interdiction cost).
4. Greedily add the highest‑ranking edge to X while the budget permits, and repeat steps 2‑4 until the budget is exhausted.

The authors evaluate the method on three classes of instances: (i) synthetic random graphs of varying density, (ii) a real‑world road network (U.S. highway system), and (iii) an Internet‑router topology. They compare against three baselines: a pure cost‑greedy interdiction (selecting cheapest edges first), a random interdiction, and exact solutions obtained via a mixed‑integer nonlinear solver on small instances. Results show that the betweenness‑based heuristic consistently yields higher expected evader costs—typically 15 % to 30 % improvements over the cost‑greedy baseline—while requiring only a fraction of the computational time (often an order of magnitude faster). Sensitivity analysis on the parameter α reveals that larger α (stronger cost bias) amplifies the benefit of targeting high‑betweenness edges, because the evader’s walk becomes more concentrated on a few low‑cost corridors that the heuristic successfully disrupts.

The paper makes three substantive contributions. First, it formalizes a realistic stochastic evader model using a Markovian walk that bridges deterministic shortest‑path behavior and fully random movement. Second, it translates this model into a non‑linear 0‑1 optimization problem, highlighting the inherent computational challenges. Third, it delivers a practical, scalable heuristic grounded in network‑science metrics, and validates its effectiveness across diverse, realistic networks. The authors suggest several avenues for future work, including extensions to multiple coordinated evaders, dynamic interdiction policies that adapt over time, and reinforcement‑learning approaches that could learn interdiction strategies directly from simulation data. Overall, the study advances both the theoretical understanding of stochastic network interdiction and provides a usable tool for practitioners facing uncertain adversarial movement.