Optimal Interdiction of Unreactive Markovian Evaders

Abstract

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. 1

Evader interdiction: algorithms, complexity and collateral damage

“…Evader e i induces a subgraph G i ⊆ G in which she roams, according to the probabilities specified by her Markov chain. An unreactive or oblivious evader [15] behaves the same regardless of the choice of sensor locations (or interdiction sites), and so her set of possible routes can be construed as objects we wish to pierce. In contrast, the reactive evader observes the locations of the sensors and can change her motion.…”

“…The node and edge settings are equivalent in general, directed graphs with location-varying costs, in the sense that a problem in one setting can be transformed into the other [15]. …”

“…The matrix M has the property that a specified target node t is a killing state: upon reaching t the evader is removed from the network. Under mild restrictions on the Markov chain (such as, t is an absorbing state), the probability of capturing the evader as a function of r can be expressed in closed form [15]: $$J(\mathbf{a},\mathbf{M},\mathbf{r})=1-{\left(\mathrm{bolda}{[\mathbf{I}-(\mathbf{M}-\mathbf{M}\odot \mathbf{r}\left)\right]}^{-1}\right)}_t$$ where the symbol ☉ indicates element-wise (Hadamard) multiplication. This formulation generalizes to a setting of multiple simultaneous evaders, each realized with probability w e , or equivalently having weight w i > 0 representing the importance of capturing her.…”

“…Any choice of some subset of nodes to observe determines a probability that a given evader will be captured (i.e., that she will pass through at least one observed node) prior to reaching her target t . The task in Budgeted Interdiction is to maximize the expected (weighted) number of evaders interdicted, subject to a budget B on sensor costs: $$\text{maximize}\sum _i{w}_i{J}_i\left(\mathbf{r}\right)\text{such that}\sum _u{r}_u{c}_u\le B$$ The special case of BI where evaders follow unreactive Markov chains is termed the Unreactive Markovian Evader (UME) interdiction problem [15]. …”

“…For brevity, we state only the key hypotheses and use the abbreviations S for stochastic, D for deterministic, FI for Full Interdiction and BI for budgeted interdiction. Node and edge interdiction often, but not always, have similar complexity [15]. …”