  This paper studies the problem of defending (1D and 2D) boundaries against a
large number of continuous attacks with a heterogeneous group of defenders. The
defender team has perfect information of the attack events within some time
(finite or infinite) horizon, with the goal of intercepting as many attacks as
possible. An attack is considered successfully intercepted if a defender is
present at the boundary location when and where the attack happens. Through
proposing a network-flow and integer programming-based method for computing
optimal solutions, and an exhaustive defender pairing heuristic method for
computing near-optimal solutions, we are able to significantly reduce the
computation burden in solving the problem in comparison to the previous state
of the art. Extensive simulation experiments confirm the effectiveness of the
algorithms. Leveraging our efficient methods, we also characterize the solution
structures, revealing the relationships between the attack interception rate
and the various problem parameters, e.g., the heterogeneity of the defenders,
attack rate, boundary topology, and the look-ahead horizon.
