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.Comment: Accepted at the Sixth International Conference on integration of AI
and OR Techniques in Constraint Programming for Combinatorial Optimization
Problems (CPAIOR 2009